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Rewind-Health Limited (New Zealand Business Number 9429046769480) was started on 09 May 2018. 2 addresses are currently in use by the company: 52 Alec Craig Way, Gulf Harbour, Whangaparaoa, 0930 (type: physical, registered). 75 Panorama Heights, Orewa, Orewa had been their physical address, up until 03 Aug 2020. Rewind-Health Limited used more aliases, namely: Global Internet Marketing Limited from 08 May 2018 to 18 Jan 2021. 100 shares are issued to 3 shareholders who belong to 2 shareholder groups. The first group includes 1 entity and holds 50 shares (50 per cent of shares), namely:
Felicity Cherry (a director) located at Gulf Harbour, Whangaparaoa postcode 0930. When considering the second group, a total of 2 shareholders hold 50 per cent of all shares (exactly 50 shares); it includes
Joseph Cherry (an individual) - located at Gulf Harbour, Whangaparaoa,
Joseph Cherry (a director) - located at Orewa, Orewa. "M694040 Internet advertising service" (ANZSIC M694040) is the category the Australian Bureau of Statistics issued Rewind-Health Limited. Our information was updated on 18 Apr 2021.
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TITLE: Application of inverse function theorem?
QUESTION [1 upvotes]: I am not completely sure if this a direct consequence of the inverse function theorem.
Assume that we have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ that we can write in terms of coordinates $x,y.$
Does the fact that $D_2f \neq 0$ mean then that we can also write $y$ as a function of $x,f$?
I feel as if my question is not completely rigorous, as $f$ is again a function depending on $x,y$ so there is somehow a circular argument here, but the question is: Assuming that I know what $x$ and $f(x,y)$ are. Does $D_2f \neq 0$ mean that I can reconstruct what $y$ was?
REPLY [1 votes]: The inverse function theorem and implicit function theorem are "cousins" of each other. You can prove one and then deduce the other. Your intuition is guiding you from the inverse function theorem towards the implicit function theorem.
Your description is slightly inaccurate (but easily fixable) in the sense that a function $f: \mathbb{R}^2 \to \mathbb{R}$ itself does not define $y$ as a function of $x$, but, the relation $f(x,y) = 0$ together with the condition $D_2(f) \neq 0$ indeed determines locally $y$ as a function of $x$. You can look up for example the book by Boothby on differentiable manifolds, or, actually, many books on differential geometry contain this.
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Page 3 of 3
Cloverleaf's demise isn't an anomaly. The new shopping centers are open and extroverted, like Stony Point and Short Pump. Willow Lawn, which enclosed a portion of the shopping center years after opening, is tearing off the roof to open up the shopping experience and create additional parking. Willow Lawn, the oldest regional shopping center in Richmond, is repositioning as an outdoor lifestyle center.
And in the last two decades, malls generally lost their stronghold on security. They were no longer seen as untouchable, safe havens for shoppers, particularly women. "In the 1960s and '70s, we started to get two-income families. The mall gave guaranteed shopping hours of 9 a.m. to 9 at night," says Gibbs, the former mall designer. "Downtowns didn't do that."
As early as 1989, Regency Square suffered an image setback when 1,500 teenagers descended on the mall the day school let out, leading to a fight and police wielding mace.
Malls coincided with the rise of modern-day suburbs, gated residential communities predicated on exclusivity and safety, and they were an extension of the ideal. Malls didn't just serve as retailing Meccas, they were de facto community gathering places, many designed to function both as a place to buy and a place to hold community meetings, put on plays and choir performances.
"A lot of people don't like them, particularly urban planners. On the other hand, they really are social centers," Gibbs says. "I think that sometimes we underestimate how much malls contribute to people's quasi-enjoyment of public spaces." Bemoan their bland design and adherence to the car, but in the suburbs, it's all that's left.
Doug Cole, an urban planner and partner at Cite Design, which Chesterfield hired to create a redevelopment plan for Cloverleaf, says there are few public spaces in the suburbs. Houses are designed with garages in the back, obscuring activity, and there are few sidewalks and public parks. He stopped by a McDonald's not long ago, and thought how odd it was that families seemed to be bringing their children primarily for the play area. "Why do you have to take your kids to McDonald's to play?" he recalls thinking.
After an expansion and a diamond-themed makeover in 1987, Chesterfield Mall was renamed Chesterfield Towne Center, an attempt to reposition the struggling mall as more upscale. It's arguable whether the mall succeeded as an upscale center, but it began drawing more customers, and retailers and power strips expanded around the mall rapidly. With the exception of Sears, the mall has long struggled to keep its anchor stores filled, seeing almost constant turnover: There was Miller & Rhoads, Thalhimers, Belk, Leggett, Hecht's, Dillard's and then J.C. Penney, which moved after closing its store at Cloverleaf store in 2000. After Hecht's, the mall added Macy's in 2006. Both Dillard's locations at the mall closed in 2008, but the mall lured Garden Ridge to take one of the anchor stores in 2010, and earlier this year added T.J. Maxx and HomeGoods to the last remaining anchor space.
Amid the decline of malls across the region, Chesterfield Towne Center has replaced all of its anchors. The mall has transitioned the southern end of the mall, facing Huguenot Road, bringing in Barnes & Noble with a shiny new outside entrance, and is planning something similar for Sears, says the mall's manager, Ashley York Venable.
Sears, for now, remains a concern. The retailer, which has more than 3,000 locations, reported $421 million in losses in the third quarter of 2011 as it struggles to compete with big-box retailers like Lowe's and Home Depot. For many malls, Sears has been the one constant.
"You wonder how long Sears can go before a major redoing," Hoffer says. "With that, you can see Sears pulling out of tons of 1960s and 1970s malls ... a horrendous retail disaster."
Chesterfield Towne Center, however, will likely survive such a scenario. It's managed to lure a healthy mix of traditional and nontraditional retailers. Earlier this year, mall management also added a farmers' market in the parking lot, open on Fridays from 9 a.m. to 2 p.m. This week the market is moving indoors, making it the first indoor farmers' market in the region. And beginning in December the indoor market will be open five days a week, Tuesday through Saturday, from 10 a.m. to 6 p.m. Vendors include bread makers, grain-fed beef butchers, produce farmers and organic egg sellers, along with several arts and crafts vendors, such as Lynne and Roger Fuller. Lynne Fuller recently started painting portraits of pets, namely dogs and a few cats. Business is brisk.
"I have all the business I need right now," Lynne Fuller says, explaining that she's currently working on 12 dog-painting orders. "I just got another order today."
Judi Williams, who manages the market, says she jumped at the opportunity to run a market that would be indoors. Once inside the mall, Williams, who also sells Indonesian coffee and teas with her husband, says the market will expand and begin offering sign language and basket-weaving classes.
"This place is mobbed," Williams says, excited about the opportunity to move inside. "You come to the mall for shoes and you remember you needed eggs. It's one-stop shopping."
Venable, the mall manager, says it's part of the destination's attempt to be more community oriented. There are two play areas inside, and the once-a-month mom's club boasts upward of 3,000 members. "We really want the mall center to become the heart of the community," she says.
Already, Chesterfield Towne Center has outlived the normal life cycle of malls, about 30 years. Venable says the mall accounts for 6 percent of the county's tax base, employs 1,650 people year round, and so far this year mall traffic is up by "double digits," Venable says. She declines to release specifics, but says the mall expects after the Christmas season, 6.5 million customers will have walked through its doors in 2011.
Whether or not the place survives as a fashion center, it represents hope. People still love the mall.
The girl at the Sunglass Hut, whom I continued dating through high school and college, still has the silky black hair and the olive cheeks. We've been married 17 years. SCorrection: In the print version of this story, Style incorrectly reported the number of members of Chesterfield Towne Center’s monthly moms club. There are 3,000 members. We regret the error.…
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\section{Simply-laced isomonodromy systems}
\label{sec:classical_systems}
Fix a finite set $J$ of cardinality $k$. Let $\pi: I \twoheadrightarrow J$ be a surjection, and write $I = \coprod_{j \in J} I^j$ for the induced partition of the finite set $I$, with parts $I^j = \pi^{-1}(j)$. Next, let $\mathcal{G}$ be the complete $k$-partite graph on nodes $I$. This means that two nodes are adjacent if and only if they lie in different parts. Both $\widetilde{\mathcal{G}}$ and $\mathcal{G}$ are by definition simply-laced, i.e. without edge loops or repeated edges. \\
Next, pick finite-dimensional complex vector spaces $\{V_i\}_{i \in I}$, as well as an embedding \begin{equation*}
a: J \hookrightarrow \comp \coprod \{\infty\}, \quad j \longmapsto a_j,
\end{equation*}
called the reading of $\widetilde{\mathcal{G}}$. The reading is said to be generic if $\infty \not\in a(J)$, and degenerate otherwise.
One also attaches complex vector spaces $W^j := \bigoplus_{i \in I^j} V_i$ to the nodes of $\widetilde{\mathcal{G}}$, simply writing $W^{\infty}$ for the (possibly nonexistent) space associated to the node $j \in J$ such that $a_j = \infty$. Finally, let us abusively denote by $\widetilde{\mathcal{G}}$ the double quiver associated to the graph $\widetilde{\mathcal{G}}$: it is the quiver on nodes $I$ having a pair of antiparallel arrows for each edge of $\widetilde{\mathcal{G}}$. The same abuse of notation will be taken for $\mathcal{G}$.
These data determine a base space of times
\begin{equation*}
\mathbf{B} := \prod_{j \in J} \comp^{I^j} \setminus \{\diags\} \subseteq \comp^I,
\end{equation*}
and a symplectic vector space
\begin{equation*}
\mathbb{M} := \bigoplus_{i \neq j \in J} \Hom\left(W^i,W^j\right),
\end{equation*}
with symplectic form
\begin{equation*}
\omega_a := \frac{1}{2}\sum_{i \neq j \in J} \Tr\big(dX^{ij} \wedge dB^{ji}\big).
\end{equation*}
Notice that $\mathbb{M}$ is the space of representations of the quiver $\widetilde{\mathcal{G}}$ with respect to the vector spaces $\{W^j\}_{j \in J}$, and one thus denotes $B^{ji}: W^i \longrightarrow W^j$ the linear maps defined by one representation; also, one defines $X^{ij}: W^j \longrightarrow W^i$ to be the scalar multiplication of $B^{ij}$ by the weight $\phi_{ij} \in \comp$, where
\begin{equation*}
\phi_{ij} = -\phi_{ji} := \begin{cases}
(a_i - a_j)^{-1}, & a_i,a_j \neq \infty \\
1, & \quad a_i = \infty
\end{cases}.
\end{equation*}
Consider now the trivial symplectic fibration $\mathbb{F}_a := (\mathbb{M},\omega_a) \times \mathbf{B} \longrightarrow \mathbf{B}$. The space $\mathbb{F}_a$ parametrises certain meromorphic connections on the trivial vector bundle
\begin{equation*}
U^{\infty} \times \comp P^1 \longrightarrow \comp P^1
\end{equation*}
with fibre $U^{\infty} := \bigoplus_{j:a_j \neq \infty} W^j$. Namely, write
\begin{equation*}
\gamma = \begin{pmatrix}
T^{\infty} & Q \\
P & B + T
\end{pmatrix}
\end{equation*}
for a generic element of $\End(W^{\infty} \oplus U^{\infty})$, where
\begin{equation*}
\Gamma := \begin{pmatrix}
0 & P \\
Q & B
\end{pmatrix} \in \mathbb{M}, \qquad \text{and} \qquad \widehat{T} := \begin{pmatrix}
T^{\infty} & 0 \\
0 & T
\end{pmatrix},
\end{equation*}
are the off-diagonal part of $\gamma$ and the diagonal of $\gamma$, respectively. One assumes that the restriction $T^j$ of $\gamma$ to $W^j$ is semisimple for all $j \in J$. Now, to a point $(\Gamma,\widehat{T}) \in \mathbb{F}_a$ one associates the connection
\begin{equation*}
\nabla = d - \mathcal{A} := d - \big(Az + (B + T) + Q(z - T^{\infty})^{-1}P\big)dz,
\end{equation*}
where
\begin{equation*}
A := \sum_{j: a_j \neq \infty} a_j\Id_j \in \End(U^{\infty}),
\end{equation*}
and $\Id_j$ is the idempotent for $W^j$. Recall the following result, from \cite{PB2012}.
\begin{thm}
\label{thm:classical_flatness}
The isomonodromy deformation (IMD) equations for the meromorphic connections above admit an Hamiltonian formulation. Moreover, the Hamiltonian system $H_i: \mathbb{F}_a \longrightarrow \comp$ is strongly flat:
\begin{equation*}
\{H_i,H_j\} = 0 = \frac{\partial H_i}{\partial t_j} - \frac{\partial H_j}{\partial t_i}, \qquad \text{for } i \neq j.
\end{equation*}
\end{thm}
Recall that the IMD equations are nonlinear first order PDEs for $\Gamma$, as a function of $\widehat{T}$. The fact that this problem admits an Hamiltonian formulation means that one can find functions $\{H_i\}_{i \in I}$ as above, such that the differential equations can be written
\begin{equation*}
\frac{\partial \Gamma_j}{\partial t_i} = \{H_i,\Gamma_j\}
\end{equation*}
for all components $\Gamma_i$ of a local section $\Gamma$ of the fibration. The definition of the Hamiltonians is implicitly given by defining the horizontal $1$-form $\varpi = \sum_{i \in I} H_i dt_i$ as
\begin{equation}
\label{eq:IMD_connection}
\varpi := \frac{1}{2}\left(\widetilde{\Xi\Gamma}\delta(\Xi\Gamma)\right) - \Tr\left(\Xi\gamma\Xi d\widehat{T}\right) + \Tr\big(X^2TdT\big) + \Tr\big(PAQ T^{\infty}dT^{\infty}\big).
\end{equation}
Here one sets $\Xi := \phi(\Gamma)$ and $X := \phi(B)$, applying the alternating weights $\phi_{ij} \in \comp$ componentwise. Also, $\delta(\Xi\Gamma)$ denotes the diagonal part of $\Xi\Gamma$ in the direct sum decomposition $W^{\infty} \oplus U^{\infty} = \bigoplus_{j \in J} W^j$, and one defines
\begin{equation*}
\widetilde{\Xi\Gamma} := \ad_{\widehat{T}}^{-1} \big[\widehat{T},\Xi\Gamma\big].
\end{equation*}
Notice that the functions $H_i$ can also be thought of as global sections of the vector bundle
\begin{equation*}
A_0 \times \mathbf{B} \longrightarrow \mathbf{B}
\end{equation*}
where $A_0 := \mathscr{O}(\mathbb{M}) \cong \Sym(\mathbb{M^*})$ is the algebra of regular function on the affine complex space $\mathbb{M}$. The Hamiltonians $H_i$ are by definition the simply-laced isomonodromy system (SLIMS). This is the classical system we wish to quantise.
\section{Potentials}
\label{sec:classical_potentials}
Consider again the complete $k$-partite quiver $\mathcal{G}$ on nodes $I = \coprod_{j \in J} I^j$.
\begin{defn}
A potential $W$ on $\mathcal{G}$ is a $\comp$-linear combination of oriented cycles in $\mathcal{G}$, defined up to cyclic permutations of their arrows. The space of potentials is denoted $\comp\mathcal{G}_{\cycl}$.
\end{defn}
Every potential $W \in \comp\mathcal{G}_{\cycl}$ defines a regular function on $\mathbb{M}$, by taking the traces of its cycles. Thus a (multi) time-dependent potential $W: \mathbf{B} \longrightarrow \comp\mathcal{G}_{\cycl}$ will define a global section $\Tr(W): \mathbf{B} \longrightarrow A_0$. Introducing the natural notation $I^i := \pi^{-1}(\pi(i)) \subseteq I$ for the part of $I$ containing the node $i \in I$, consider the following potentials:
\begin{equation}
\label{eq:classical_potentials}
\begin{split}
W_i(2) &:= \sum_{j \in I \setminus I^i} (t_i - t_j) \alpha_{ij}\alpha_{ji} \\
W_i(3) &:= \sum_{j,l \in I \setminus I^i:I^j \neq I^l} (a_j - a_l)\alpha_{il}\alpha_{lj}\alpha_{ji} \\
W_i(4) &:= \sum_{m \in I^i \setminus \{i\},j,l \in I \setminus I^i} (a_i - a_j)(a_i - a_l)\frac{\alpha_{ij}\alpha_{jm}\alpha_{ml}\alpha_{li}}{t_i - t_m}
\end{split}
\end{equation}
where $\alpha_{ij}$ is the arrow from $j \in I$ to $i \in I$ in $\mathcal{G}$. We agree to write a cycle in $\mathcal{G}$ as the sequence of its arrows, reading from right to left.
\begin{prop}
The Hamiltonian $H_i$ of the simply-laced isomonodromy system is the sum of the traces of these potentials, for a generic reading of $\mathcal{G}$:
\begin{equation*}
H_i = \Tr(W_i(4)) + \Tr(W_i(3)) + \Tr(W_i(2)).
\end{equation*}
Moreover, in a degenerate reading one only needs to change the weights of the same types of cycles.
\end{prop}
\begin{proof}
It follows from an explicit expansion of the formula for $\varpi$.
\end{proof}
This in particular implies that the flatness does not depend in a crucial way on whether the reading is generic or not. Let us now introduce some terminology, for further use.
\begin{defn}
The potentials $W_i(n)$ above are called the (classical) IMD potentials, for $i \in I, 2 \leq n \leq 4$. Their addends will be referred to as the IMD cycles. The IMD $4$-cycles can be further divided in two families:
\begin{enumerate}
\item nondegenerate, if they touch $4$ distinct nodes of $\mathcal{G}$.
\item degenerate, if they touch $3$ distinct nodes of $\mathcal{G}$.
\end{enumerate}
\end{defn}
This provides the following types of cycles:
\begin{center}
\begin{tikzpicture}
\vertex (a) at (0,0) {};
\vertex (b) at (2,0) {};
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b);
\end{tikzpicture} \qquad \begin{tikzpicture}
\vertex (a) at (0,0) {};
\vertex (b) at (2,0) {};
\vertex (c) at (1,1.7172) {};
\path
(a) edge [->, line width = 1.3] (b)
(b) edge [->, line width = 1.3] (c)
(c) edge [->, line width = 1.3] (a);
\end{tikzpicture} \qquad \begin{tikzpicture}
\vertex (a) at (0,0) {};
\vertex (b) at (2,0) {};
\vertex (c) at (2,2) {};
\vertex (d) at (0,2) {};
\path
(a) edge [->, line width = 1.3] (b)
(b) edge [->, line width = 1.3] (c)
(c) edge [->, line width = 1.3] (d)
(d) edge [->, line width = 1.3] (a);
\end{tikzpicture} \qquad
\begin{tikzpicture}
\vertex(a) at (0,0) {};
\vertex (b) at (2,0) {};
\vertex (c) at (1,1.7172) {};
\path
(a) edge [->, line width = 1.3, bend left = 20] (c)
(a) edge [<-, line width = 1.3, bend right = 20] (c)
(b) edge [->, line width = 1.3, bend left = 20] (c)
(b) edge [<-, line width = 1.3, bend right = 20] (c);
\end{tikzpicture}
\end{center}
In order from left to right, one has $2$-cycles, $3$-cycles, nondegenerate $4$-cycles and degenerate $4$-cycles. Notice that the degenerate $4$-cycle are the glueing of two $2$-cycles at a node, that will be called their centre. The other two nodes must lie in one and the same part of $I$.
There is also an intrinsic way to think of traces. Namely, if $C = \alpha_n \dots \alpha_1$ is an oriented cycle in $\mathcal{G}$ starting at a node $i \in I$, and for all representation $\rho \in \mathbb{M}$ of $\mathcal{G}$, one gets an endomorphism $\rho^C = \rho^{\alpha_n} \circ \dots \circ \rho^{\alpha_1}$ of $V_i$. However, this object can also be thought as living in $A_0 \otimes \End(V_i)$, since all its components define regular functions on $\mathbb{M}$. Now taking traces amounts to contract $V_i$ and $V_i^*$, leaving a function $\Tr(C) = \Tr(\rho^C) \in A_0$.
As a last remark, there is a natural (positively) graded Lie structure $\{,\}$ on $\comp\mathcal{G}_{\cycl}$, where the gradation is given by cycle length, called the necklace Lie algebra structure (see e.g. \cite{BL-B2002, PE2007}). We shall call $C \in \comp\mathcal{G}_{\cycl}$ an $m$-cycle if it has $m$ arrows, and also set $l(C) := m$ in that case. Also, if $\alpha$ is an arrow in $\mathcal{G}$, we write $\alpha^*$ for its (unique) antiparallel one.
\begin{defn}
Pick two oriented cycles $C_1 = \alpha_n \dots \alpha_1$ and $C_2 = \beta_m \dots \beta_1$ in $\mathcal{G}$. The Lie bracket $\{C_1,C_2\}$ is a weighted sum of $(n+m-2)$-cycles obtained as follows. For all pairs of antiparallel arrows $\alpha_i, \beta_j = \alpha_i^*$, one deletes that pair and glues together the two remaining cycles. \\
The weights are determined by the defining relation of the Poisson bracket of $A_0$.
\end{defn}
To see this graphically, fix a pair $i,j$ such that $\alpha_i = \beta_j^*$, and introduce the notation $t(\alpha), h(\alpha) \in I$ for the tail and the head of an arrow $\alpha$ in $\mathcal{G}$, respectively; these are the starting node of $\alpha$ and the end node of $\alpha$, respectively. Set then $a = t(\beta_{j-1}), b = h(\beta_{j-1}) = h(\alpha_i), c = h(\beta_j) = h(\alpha_{i-1}), d = h(\beta_{j+1}), e = t(\alpha_{i-1}), f = h(\alpha_{i+1}) \in I$. Then the local picture before deleting arrows looks like this:
\begin{center}
\begin{tikzpicture}
\node (a) at (0,0.5) {};
\vertex (b) at (2,0.5) [label = below:$a$] {};
\vertex (c) at (2,2) [label = left:$b$] {};
\vertex (d) at (4,2) [label = right:$c$] {};
\vertex (e) at (4,0.5) [label = below:$d$] {};
\node (f) at (6,0.5) {};
\node (g) at (7,3) {};
\vertex (h) at (6,2) [label = right:$e$] {};
\vertex (i) at (4,2.5) [label = above:$c$] {};
\vertex (j) at (2,2.5) [label = above:$b$] {};
\vertex (k) at (0,2) [label = left:$f$] {};
\node (l) at (-1,3) {};
\node (m) at (3,2) [label = below:$\beta_j$] {};
\node (n) at (3,2.5) [label = above:$\alpha_i$] {};
\node (o) at (2,1.3) [label = left:$\beta_{j-1}$] {};
\node (p) at (4,1.3) [label = right:$\beta_{j+1}$] {};
\node (q) at (5.2,2.25) [label = above:$\alpha_{i-1}$] {};
\node (r) at (0.8,2.25) [label = above:$\alpha_{i-1}$] {};
\path
(a) edge [dashed] (b)
(b) edge [->,line width = 1.3] (c)
(c) edge [->,line width = 1.3] (d)
(d) edge [->,line width = 1.3] (e)
(e) edge [dashed] (f)
(g) edge [dashed] (h)
(h) edge [->,line width = 1.3] (i)
(i) edge [->, line width = 1.3] (j)
(j) edge [->,line width = 1.3] (k)
(k) edge [dashed] (l);
\end{tikzpicture}
\end{center}
Afterwards, one will have:
\begin{center}
\begin{tikzpicture}
\node (a) at (-1,0.5) {};
\vertex (b) at (1,0.5) [label = below:$a$] {};
\vertex (c) at (1,2.5) [label = above:$b$] {};
\vertex (d) at (-1,2) [label = left:$f$] {};
\node (e) at (-2,3) {};
\node (f) at (6,3) {};
\vertex (g) at (5,2) [label = right:$e$] {};
\vertex (h) at (3,2.5) [label = above:$c$] {};
\vertex (i) at (3,0.5) [label = below:$d$] {};
\node (j) at (5,0.5) {};
\node (k) at (1,1.3) [label = left:$\beta_{j-1}$] {};
\node (l) at (3,1.3) [label = right:$\beta_{j+1}$] {};
\node (m) at (4.2,2.25) [label = above:$\alpha_{i-1}$] {};
\node (n) at (-0.2,2.25) [label = above:$\alpha_{i-1}$] {};
\path
(a) edge [dashed] (b)
(b) edge [->,line width = 1.3] (c)
(c) edge [->,line width = 1.3] (d)
(d) edge [dashed] (e)
(f) edge [dashed] (g)
(g) edge [->,line width = 1.3] (h)
(h) edge [->,line width = 1.3] (i)
(i) edge [dashed] (j);
\end{tikzpicture}
\end{center}
Now, the nice fact is that this bracket comes from the Poisson structure of $A_0$.
\begin{prop}
\label{prop:classicalbrackets}
One has \begin{equation*}
\Tr\big\{C_1,C_2\} = \{\Tr(C_1),\Tr(C_2)\} \in A_0
\end{equation*}
for all cycles $C_1,C_2 \in \comp\mathcal{G}_{\cycl}$.
\end{prop}
The proof consists of a direct expansion of the Poisson bracket
\begin{equation*}
\{\Tr(C_1),\Tr(C_2)\} = \{\Tr(X^{\alpha_n} \dots X^{\alpha_1}),\Tr(X^{\beta_m} \dots X^{\beta_1})\},
\end{equation*}
which will be provided in the appendix \S~\ref{sec:appendix}. Conceptually, however, what happens is the following. The invariant regular functions on $\mathbb{M}$ for the action of $G := \prod_{j \in J} \GL_{\comp}(W^j)$ consist of the $\comp$-algebra $A_0^G \subseteq A_0$ generated by traces of cycles. Hence we have an injective map $\Tr: \comp\mathcal{G}_{\cycl} \hookrightarrow A_0^G$, and the above discussion shows that this is a Lie algebras' morphism: the necklace Lie bracket is the pull-back of the Poisson bracket on $A_0$.
Last, notice that it is not possible to upgrade $\comp\mathcal{G}_{\cycl}$ to a Poisson algebra using the natural concatenation product, since $\Tr(AB) \neq \Tr(A)\Tr(B)$ for general endomorphisms $A,B$ of a vector space. Rather, one should define a formal product of cycles that satisfies the same rules as the product of their traces, i.e. be commutative. This is well expressed by the following elementary algebraic fact.
\begin{prop}
\label{prop:algebraictrick}
Pick a complex vector space $A$ together with a linear embedding $\iota: V \hookrightarrow B$ into a $\comp$-algebra. Then there is a natural tensor map $\Tens(\iota): \Tens(V) \longrightarrow B$, defined on pure tensors as
\begin{equation*}
\Tens(\iota)(v_1 \otimes \dots \otimes v_n) := \iota_1(v) \dots \iota_n(v) \in B.
\end{equation*}
This map is surjective on the subalgebra $B.\iota(V) \subseteq B$ generated by the image of $\iota$ in $B$, and it induces an isomorphism of algebras
\begin{equation*}
\Tens(V) / \Ker(\Tens(\iota)) \cong B.\iota(V).
\end{equation*}
\end{prop}
This is an application of the universal properties of tensor products and quotients. In the case at hand, one just finds
\begin{equation*}
A^G_0 \cong \Tens(\comp\mathcal{G}_{\cycl}) / \Ker(\Tens(\Tr)) = \Sym(\comp\mathcal{G}_{\cycl}),
\end{equation*}
so that it makes sense to define
\begin{equation*}
A^{\mathcal{G}}_0 := \Sym(\comp\mathcal{G}_{\cycl}).
\end{equation*}
The identification $A^G_0 \cong A^{\mathcal{G}}_0$ is just saying that all $G$-invariant regular functions on $\mathbb{M}$ are monomials of (traces of) cycles, with commutative variables. Notice that the Lie bracket of $\comp\mathcal{G}_{\cycl}$ is now tautologically upgraded to a Poisson bracket, and $A_0^{\mathcal{G}}$ is isomorphic to $A_0^G$ as a graded commutative Poisson algebra. We shall present a quantum counterpart of this, in \S~\ref{sec:quantisation_potentials}.
As an application of this cycle-theoretic viewpoint, one can provide a direct proof of ``half'' of the strong flatness of the SLIMS. More precisely, remark that one has
\begin{equation*}
\Tr(\partial_{t_i}W_j) = \partial_{t_i}\Tr(W_j)
\end{equation*}
for all $i,j \in I$, where $W_j$ is an IMD potential. This is because the derivative does not modify the cycles that make up the potentials, but only their weights. Hence showing that $\partial_{t_i}H_j - \partial_{t_j}H_i = 0$ is equivalent to showing that $\partial_{t_i}W_j - \partial_{t_j}W_i = 0$, because of the injectivity of $\Tr: \comp\mathcal{G}_{\cycl} \hookrightarrow A_0$.
\begin{prop}
\label{prop:direct_verification}
One has $\partial_{t_i}W_j = \partial_{t_j}W_i$ for all $i,j \in I$.
\end{prop}
\begin{proof}
One can clearly assume $i \neq j \in I$. Then one has
\begin{equation*}
\partial_{t_j}W_i(2) = \begin{cases}
-\alpha_{ij}\alpha_{ji}, & I^i \neq I^j \\
0, & \text{else}
\end{cases} \qquad \text{and} \qquad \partial_{t_i}W_j(2) = \begin{cases}
-\alpha_{ji}\alpha_{ij}, & I^i \neq I^j \\
0, & \text{else}
\end{cases}.
\end{equation*}
Also
\begin{equation*}
\partial_{t_j}{W_i}(3) = 0 = \partial_{t_i}(W_j),
\end{equation*}
since all $3$-cycles are actually time-independent in our setting. Finally,
\begin{equation*}
\partial_{t_j}W_i(4) = \begin{cases}
\sum_{m,l \in I \setminus I^i} (a_i - a_m)(a_i - a_l) \frac{\alpha_{im}\alpha_{mj}\alpha_{jl}\alpha_{li}}{(t_i - t_j)^2}, & I^i = I^j \\
0, & \text{else}
\end{cases}
\end{equation*}
and
\begin{equation*}
\partial_{t_i}W_j(4) = \begin{cases}
\sum_{m,l \in I \setminus I^j} (a_j - a_m)(a_j - a_l) \frac{\alpha_{jl}\alpha_{li}\alpha_{im}\alpha_{mj}}{(t_j - t_i)^2}, & I^i = I^j \\
0, & \text{else}
\end{cases}.
\end{equation*}
This is seen explicitly on the formulas \eqref{eq:classical_potentials}, and proves the claim, because
\begin{equation*}
\alpha_{ij}\alpha_{ji} = \alpha_{ji}\alpha_{ij} \in \comp\mathcal{G}_{\cycl} \qquad \text{and} \qquad \alpha_{jl}\alpha_{li}\alpha_{im}\alpha_{mj} = \alpha_{im}\alpha_{mj}\alpha_{jl}\alpha_{li} \in \comp\mathcal{G}_{\cycl}.
\end{equation*}
For the case of $4$-cycles, one must also recall that $a_i = a_j$ if $I^i = I^j$, because the reading only depends on the parts of $I$.
\end{proof}
\section{Quantisation: algebras}
\label{sec:quantisation_algebras}
Consider again the commutative Poisson algebra $A_0 = \mathscr{O}(\mathbb{M}) \cong \Sym(\mathbb{M}^*)$ of regular functions $\mathbb{M}$.
\begin{defn}
A one-parameter deformation quantisation of $A_0$ is a topologically free $\comp[[\hbar]]$-algebra $\widehat{A}$, together with an identification $\widehat{A} / \hbar \widehat{A} \cong A_0$, such that the Poisson bracket $\{\cdot,\cdot\}$ of $A_0$ is naturally induced by the commutator $\big[\cdot,\cdot\big]$ of $\widehat{A}$:
\begin{equation*}
\{x,y\} = \frac{1}{\hbar} \big[ \widehat{x},\widehat{y}\big] + O(\hbar),
\end{equation*}
where $x,y \in A_0, \widehat{x},\widehat{y} \in \widehat{A}$ are arbitrary lifts, and the analytical Landau notation $O(\hbar)$ stands for an arbitrary element of the ideal $\hbar \widehat{A} \subseteq \widehat{A}$ generated by $\hbar$.
\end{defn}
In the case we consider one can avoid using $\comp[[\hbar]]$-modules, by performing filtered quantisation (see \cite{PE2007}).
\subsection{The Weyl algebra}
\label{sec:Weyl}
\ \\
There exists an associative (noncommutative) filtered algebra $A$ whose associated graded is canonically isomorphic to $A_0$. Moreover, if one denotes $\sigma: A \longrightarrow \gr(A) \cong A_0$ the grading map, then one has
\begin{equation}
\label{eq:filtered_quantisation}
\sigma \big[\widehat{x},\widehat{y}\big] = \{x,y\},
\end{equation}
for $x,y \in A_0$ and for any lift $\widehat{x}, \widehat{y} \in A$. The algebra $A$ is defined as follows.
\begin{defn}
Set
\begin{equation*}
A = W(\mathbb{M},\omega_a) := \Tens(\mathbb{M}) / I_{\omega_a},
\end{equation*}
where $\Tens(\mathbb{M})$ is the tensor algebra of the vector space $\mathbb{M}$, and $I_{\omega_a} \subseteq \Tens(\mathbb{M})$ is the two-sided ideal generated by elements of the form
\begin{equation*}
x \otimes y - y \otimes x - \omega_a(x,y),
\end{equation*}
for $x,y \in \mathbb{M}$. This is the Weyl algebra of the symplectic vector space $(\mathbb{M},\omega_a)$.
\end{defn}
The filtration of $A$ is the quotient filtration induced by the natural filtration of $\Tens(\mathbb{M})$. Notice that there is a canonical linear isomorphism $\varphi: \mathbb{M} \longrightarrow \mathbb{M}^*$, induced by the nondegenerate pairing $\mathbb{M} \wedge \mathbb{M} \longrightarrow \comp$ provided by the symplectic form $\omega_a$. Moreover, there is a unique symplectic structure on $\mathbb{M}^*$ such that $\varphi$ is a symplectomorphism, which we abusively denote $\omega_a$ as well. The Weyl algebra $W(\mathbb{M}^*,\omega_a)$ of the dual symplectic space is then canonically isomorphic to $A$, and it is not really necessary to distinguish the two, as far as generator and relations are concerned.\footnote{The canonical isomorphism is given by the universal property of the quotient applied to the composition $\pi \circ \Tens(\varphi): \Tens(\mathbb{M}) \longrightarrow W(\mathbb{M}^*,\omega_a)$, where $\Tens(\varphi): \Tens(\mathbb{M}) \longrightarrow \Tens(\mathbb{M}^*)$ is the image of $\varphi$ under the functor $\Tens$, and $\pi: \mathbb{M}^* \longrightarrow W(\mathbb{M}^*,\omega_a)$ is the canonical projection.}~The intrinsic way of thinking about this is the following. The symplectic vector space $(\mathbb{M},\omega_a)$ is equipped with a Poisson bracket $\{\cdot,\cdot\}: \mathscr{O}(\mathbb{M}) \wedge \mathscr{O}(\mathbb{M}) \longrightarrow \mathscr{O}(\mathbb{M})$ such that the degree of the polynomial function $\{f,g\}$ equals $\deg(f) + \deg(g) - 2$, for $f,g \in \mathscr{O}(\mathbb{M})$. In particular, its restriction to linear functions yields an alternating bilinear map $\{\cdot,\cdot\}: \mathbb{M}^* \wedge \mathbb{M}^* \longrightarrow \comp$. Thus one may say that the Weyl algebra is obtained from the tensor algebra by modding out the Poisson structure, just as for the universal enveloping algebras $U(\liealg)$ of a Lie algebra $\liealg$.
Finally, consider the composition $\sigma \circ \pi: \Tens(\mathbb{M}^*) \longrightarrow \gr(A)$ of the canonical projection $\pi: \Tens(\mathbb{M}^*) \longrightarrow A$ with the grading map $\sigma: A \longrightarrow \gr(A)$. One may show that this vanishes precisely on the homogeneous two-sided ideal $I_0 \subseteq \Tens(\mathbb{M}^*)$ generated by commutators, thus inducing an isomorphism
\begin{equation*}
\gr(A) \cong \Tens(\mathbb{M}^*) / I_0 = \Sym(\mathbb{M}^*) \cong A_0.
\end{equation*}
The quantisation identity \eqref{eq:filtered_quantisation} can be shown by a direct inspection on elements of order one, which generate $A_0$. There is now a universal way of reconstructing a $\hbar$-deformation quantisation of $A_0$ from $A$.
\subsection{Rees construction}
\label{sec:Rees}
\ \\
Consider an associative positively filtered algebra $B = \bigcup_{k \geq 0} B_{\leq k}$. Recall that one calls
\begin{equation*}
\ord(b) := \min \{k \geq 0 \mid b \in B_{\leq k}\}
\end{equation*}
the order of the element $b \in B$.
\begin{defn}
The Rees algebra $\Rees(B)$ of $B$ is the $\comp[\hbar]$-algebra defined by
\begin{equation*}
\Rees(B) := \bigoplus_{k \geq 0} B_{\leq k} \hbar^k \subseteq B[\hbar].
\end{equation*}
\end{defn}
\begin{prop}
There exists a topologically-free $\comp[[\hbar]]$-algebra $\widehat{A}$ that defines a deformation quantisation of $A_0$. It is obtained via (a completion of) the Rees algebra of $A$.
\end{prop}
\begin{proof}
It suffices to set:
\begin{equation*}
\widehat{A} := \left\lbrace \left.\sum_{k \geq 0} d_k\hbar^k \right| d_k \in A_{\leq k} \text{ for all } k \geq 0, \lim_{k \longrightarrow +\infty} (k - \ord(d_k)) = +\infty\right\rbrace \subseteq A[[\hbar]].
\end{equation*}
One can indeed show that the map
\begin{equation*}
\varphi: \sum_{k \geq 0} d_k \hbar^k \longmapsto \sum_{k \geq 0} \sigma_k(d_k)
\end{equation*}
induces a (canonical) isomorphism $\widehat{A} / \hbar\widehat{A} \cong A_0$. Here $\sigma_k: A_{\leq k} \longrightarrow A_{\leq k} / A_{\leq k-1} \cong (A_0)_k$ is the order $k$ part of the grading map, that is the canonical projection. Moreover, the identity of the relevant Poisson brackets follows from \eqref{eq:filtered_quantisation}.
\end{proof}
Thanks to this universal construction, it makes sense to speak of the Weyl algebra $A$ as a quantisation of $A_0$. This is the first step to actually quantise observables $f \in A_0$, that is to provide a map $f \longmapsto \widehat{f}$ that is a section of the semiclassical limit $\sigma$ (among other properties which are not crucial to list at present). Nonetheless, there is a natural way to quantise elements of order $1$, i.e. linear functions on $\mathbb{M}$. Namely, one considers the composition $\pi \circ \iota: \mathbb{M}^* \longrightarrow \Tens(\mathbb{M}^*) \longrightarrow A$ of the canonical embedding $\iota: \mathbb{M}^* \hookrightarrow \Tens(\mathbb{M}^*)$ with the canonical projection $\pi: \Tens(\mathbb{M}^*) \longrightarrow A$, and sets $\widehat{f} := \pi (\iota (f))$ for all linear functions $f: \mathbb{M} \longrightarrow \comp$. This is well defined because the ideal $I_{\omega_a}$ defining the Weyl algebra does not intersect the space $\Tens(\mathbb{M}^*)_{\leq 1} = \comp \oplus \mathbb{M}^*$.
\section{Quantisation: potentials}
\label{sec:quantisation_potentials}
Just as we coded (invariant) functions on $\mathbb{M}$ via cycles, we can code quantum operators via decorated cycles. Consider again a complete $k$-partite quiver $\mathcal{G}$.
\begin{defn}
An anchored cycle $\widehat{C}$ is an oriented cycle in $\mathcal{G}$ with a starting arrow fixed, to be called the anchor of $\widehat{C}$. We will denote this by underlining the anchor:
\begin{equation*}
\widehat{C} = \alpha_n \dots \underline{\alpha_1},
\end{equation*}
where $\alpha_n, \dots, \alpha_1$ are arrows in $\mathcal{G}$.
\end{defn}
The idea is the following. Using the above ``linear quantisation'' $X \longmapsto \widehat{X}: A_0 \longrightarrow A$ one can now associate $n!$ different quantum operators to all monomial $f = X_1 \dots X_n$ of degree $n$; namely, one has
\begin{equation*}
\widehat{X}_{\sigma(1)} \dots \widehat{X}_{\sigma(n)} \in A_{\leq n},
\end{equation*}
for all permutation $\sigma \in \Sigma_n$ on $n$ objects. There is in general no canonical way to pick one of them. This is the main issue of quantisation: extending the quantisation $X \longmapsto \widehat{X}$ of linear functions to a full quantisation $f \longmapsto \widehat{f}: A_0 \longrightarrow A$.
Nevertheless, suppose that $f \in A_0$ is a monomial coming from the trace of a cycle $C = \alpha_n \dots \alpha_1 \in \comp\mathcal{G}_{\cycl}$. This means that $f$ is one monomial of the sum
\begin{equation*}
\Tr(C) = \sum_K X^{\alpha_n}_{k_n,k_{n-1}} \dots X^{\alpha_1}_{k_1,k_n} \in A_0,
\end{equation*}
where $K = (k_n,\dots,k_1)$ is an appropriate multi-index. Now, if one picks an anchor for $C$, say that $\widehat{C} := \alpha_n \dots \underline{\alpha_1}$, then the quantum operator
\begin{equation*}
\widehat{f} := \sum_{K \in D_C} \widehat{X}^{\alpha_n}_{k_n,k_{n-1}} \dots \widehat{X}^{\alpha_1}_{k_1,k_n} \in A_{\leq n}
\end{equation*}
is uniquely defined, and one can in turn define $\Tr(\widehat{C}) \in A$ to be that operator. In hindsight, and more intrinsecally, one could just consider the operator-valued matrix
\begin{equation*}
\widehat{\rho}^{\widehat{C}} := \widehat{X}^{\alpha_n} \dots \widehat{X}^{\alpha_1} \in A \otimes \End(V_i),
\end{equation*}
where $i := t(\alpha_1) \in I$ is the starting node of $\widehat{C}$. Taking trace then amounts again to contracting $V_i$ against $V_i^*$.
As a final remark, notice that two different anchored cycles $\widehat{C}_1, \widehat{C}_2$ may define the same quantum operator. This happens when their two underlying cycles coincide under an ``admissible'' permutation of their arrows: no arrow $\alpha$ may pass over its antiparallel $\alpha^*$. This is because the entries of matrices $\widehat{X}^{\alpha}, \widehat{X}^{\beta}$ commute if and only if $\alpha \neq \beta^*$, according to the defining relations of the Weyl algebra. This motivates the next definitions.
\begin{defn}
Consider an anchored cycle $\widehat{C} = \alpha_n \dots \underline{\alpha_1}$ on $\mathcal{G}$. An admissible permutation of its arrows consists in dividing the word $\alpha_n \dots \alpha_1$ in two subwords
\begin{equation*}
A = \alpha_n \dots \alpha_{n-i}, \qquad B = \alpha_{n-i-1} \dots \alpha_1
\end{equation*}
such that no arrow in $A$ has its antiparallel in $B$, and to swap $A$ and $B$. This yields a new anchored cycle $\widehat{C}' = \alpha_{n-i-1} \dots \alpha_1 \alpha_n \dots \underline{\alpha_i}$.
\end{defn}
\begin{defn}
Let $\widehat{\comp\mathcal{G}_{\cycl}}$ be the complex vector space spanned by anchored cycles in $\mathcal{G}$, defined up to admissible permutations of their arrows. Its elements will be called quantum potentials, its generators quantum cycles. \\
One denotes by $\sigma: \widehat{\comp\mathcal{G}_{\cycl}} \longrightarrow \comp\mathcal{G}$ the map that forgets the anchor, which we call again the semiclassical limit. A quantum potential $\widehat{W}$ is a quantisation of the potential $W$ if $\sigma(\widehat{W}) = W$.
\end{defn}
There exists now a well defined $\comp$-linear injective map $\Tr: \widehat{\comp\mathcal{G}_{\cycl}} \hookrightarrow A$, together with a commutative square where the quantum and classical traces intertwine the semiclassical limit: $\sigma\big(\Tr(\widehat{C})\big) = \Tr(C) \in A_0$ for all quantum cycles $\widehat{C} \in \widehat{\comp \mathcal{G}_{\cycl}}$ that quantises the classical cycle $C \in \comp\mathcal{G}_{\cycl}$. Moreover, one can use Prop.~\ref{prop:algebraictrick} to produce a cycle-theoretic analogue of the Weyl algebra. Namely, one considers the tensor map
\begin{equation*}
\Tens(\Tr): \Tens(\widehat{\comp\mathcal{G}_{\cycl}}) \longrightarrow A,
\end{equation*}
which is surjective on the subalgebra $A^G \subseteq A$ of $A$, which is by definition the subalgebra generated by traces of quantum cycles. One thus has an isomorphism of associative algebras
\begin{equation*}
\Tens(\widehat{\comp\mathcal{G}_{\cycl}}) / \Ker(\Tens(\Tr)) \cong A^G.
\end{equation*}
Now, setting $A^{\mathcal{G}}$ to be the quotient on the left-hand side, one has constructed an associative (quantum) algebra that has an analogous relation with $A^G$ as $A^{\mathcal{G}}_0$ has with $A^G_0$. Notice that this is abstract, as we do not have a nice description of the kernel of the quantum trace map. However, one still has an identification
\begin{equation*}
\comp \oplus \widehat{\comp\mathcal{G}_{\cycl}} \cong A^{\mathcal{G}}_{\leq 1},
\end{equation*}
with respect to the quotient filtration on $A^{\mathcal{G}}$. Indeed, this happens because $\Tens(\Tr)$ is by definition the identity on $\comp$ (trace of empty cycles, if one will), and it is injective on the vector space generated by quantum cycles.
Finally, notice that $A^{\mathcal{G}}$ now has a well defined product, defined on quantum cycles by
\begin{equation*}
\Tr\big(\widehat{C}_1\widehat{C}_2\big) = \Tr(\widehat{C_1})\Tr(\widehat{C}_2) \in A.
\end{equation*}
This is basically a $\star$-product, deforming the commutative one of $A_0^{\mathcal{G}}$. This makes $A^{\mathcal{G}}$ into a filtered associative algebra, still provided with a semiclassical limit
\begin{equation*}
\sigma: A^{\mathcal{G}} \longrightarrow A_0^{\mathcal{G}},
\end{equation*}
which is defined on monomials by forgetting anchors $\sigma: \widehat{C}_1 \dots \widehat{C}_n \longmapsto \sigma(\widehat{C}_1) \dots \sigma(\widehat{C}_n)$.
In this noncommutative context it is even more important to allow for formal products of cycles, in order to keep track of the anchoring, as exemplified by the next proposition.
\begin{prop}
\label{prop:change_anchor}
Pick two quantum cycles $\widehat{C},\widehat{C}'$ such that their underlying classical cycles $\sigma(\widehat{C}), \sigma(\widehat{C}')$ coincide. Then their difference is a sum of products of pairs of quantum cycles whose lengths sum to $l(\widehat{C}) - 2 = l(\widehat{C}') - 2$.
\end{prop}
A proof of this is given in \S~\ref{sec:appendix}.
\section{Simply-laced quantum connection}
\label{sec:quantum_connections}
The following few definitions now come naturally. Consider again the classical IMD cycles of \S~\ref{sec:classical_potentials}. The $3$-cycles and the nondegenerate $4$-cycles do not contain pairs of antiparallel arrows, so that one can quantise such a cycle $C$ by choosing any anchor: all of them are equivalent. As for $2$-cycles and degenerate $4$-cycles, one makes the following choices.
\begin{defn}
The quantisation of a degenerate $4$-cycles is the quantum cycle having the same underlying classical cycle, anchored at any arrow coming out of its centre. The quantisation of a two cycle $C = $~\begin{tikzpicture}
\vertex (a) at (0,0) {};
\vertex (b) at (2,0) {};
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b);
\end{tikzpicture}
is by definition the quantum potential
\begin{center}
$\widehat{C} = \frac{1}{2}\Bigg($\begin{tikzpicture}
\vertex (a) at (0,0) [fill = black] {};
\vertex (b) at (2,0) {};
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b);
\end{tikzpicture} $+$ \begin{tikzpicture}
\vertex (a) at (0,0) {};
\vertex (b) at (2,0) [fill = black] {};
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b);
\end{tikzpicture}$\Bigg)$.
\end{center}
\end{defn}
In this picture and in all that follow, the black nodes are the tail of the anchor. As for the degenerate $4$-cycles, a priori specifying a starting arrow is more than specifying a starting node, but in this case there is no ambiguity: changing the order of the arrows coming out of the central node amounts to an admissible permutation of the arrows of the degenerate $4$-cycles. This is because such a cycle can be written as a word $C = \beta^*\beta\alpha^*\alpha$, where $\alpha,\beta$ are the two distinct arrows of $\mathcal{G}$ coming out of the centre. Now, the two possible anchors at the centre correspond to the quantisations $\widehat{C}_1 = \beta^*\beta\alpha^*\underline{\alpha}$ and $\widehat{C}_2 = \alpha^*\alpha\beta^*\underline{\beta}$. These two are seen to be equal, by means of the cyclical permutation that swaps the two $2$-cycles: one can move $\beta^*\beta$ to the right of $\alpha^*\alpha$ without changing the relative order of the antiparallel pairs $(\alpha,\alpha^*), (\beta,\beta^*)$.
This is totally canonical, and does not rely on a full quantisation $\mathcal{Q}: A_0 \longrightarrow A$. One can however show that it amounts to correcting the standard Weyl quantisation. Consider now the IMD potentials $W_i = W_i(4) + W_i(3) + W_i(2) \in \comp\mathcal{G}_{\cycl}$ of \eqref{eq:classical_potentials}.
\begin{defn}
\label{def:quantum_Hamiltonians}
The quantum IMD potential $\widehat{W}_i \in \widehat{\comp \mathcal{G}_{\cycl}}$ at the node $i$ is the sum of the quantisations of its IMD cycles. The quantum IMD Hamiltonian $\widehat{H}_i: \mathbf{B} \longrightarrow A$ is the trace of the quantum IMD potential at the node $i$: $\widehat{H}_i := \Tr(\widehat{W}_i)$.
\end{defn}
This is a quantisation of the classical IMD Hamiltonian $H_i: \mathbf{B} \longrightarrow A_0$, in the sense that the identity $\sigma(\widehat{H}_i) = H_i$ is true everywhere on $\mathbf{B}$. Consider now the trivial bundle $\mathbb{E}_a := A \times \mathbf{B} \longrightarrow \mathbf{B}$.
\begin{defn}
The (universal) simply-laced quantum connection (SLQC) $\widehat{\nabla}$ is the connection on $\mathbb{E}_a$ defined by
\begin{equation*}
\widehat{\nabla} := d_{\mathbf{B}} - \widehat{\varpi}, \qquad \text{where } \widehat{\varpi} := \sum_{i \in I} \widehat{H}_i dt_i.
\end{equation*}
\end{defn}
Note that $\Omega^1(\mathbf{B},A) \subseteq \Omega^1\big(\mathbf{B},\End(\mathbb{E}_a)\big)$, where one lets $\widehat{H}_i$ act linearly on the fibre $A$ of $\mathbb{E}_a$ by left multiplication. The main result is the following.
\begin{thm}
\label{thm:quantum_flatness}
$\widehat{\nabla}$ is strongly flat, i.e.
\begin{equation*}
\big[\widehat{H}_i,\widehat{H}_j\big] = 0 = \frac{\partial \widehat{H}_i}{\partial t_j} - \frac{\partial \widehat{H}_j}{\partial t_i}, \qquad \text{for } i,j \in I.
\end{equation*}
\end{thm}
\section{Proof of strong flatness: I}
\label{sec:flatness_I}
Let us show that
\begin{equation*}
\partial_{t_i}\widehat{H}_j - \partial_{t_j}\widehat{H}_i = 0
\end{equation*}
for all $i,j \in I$. This follows from a lemma.
\begin{lem}
Pick a classical IMD potential $W_i: \mathbf{B} \longrightarrow \comp\mathcal{G}$. Then
\begin{equation*}
\partial_{t_j}\widehat{W}_i = \widehat{\partial_{t_j}W_i}, \qquad \partial_{t_j}\Tr(\widehat{W}_i) = \Tr(\partial_{t_j}\widehat{W}_i),
\end{equation*}
for all $j \in I$.
\end{lem}
\begin{proof}
The first set of identities are due to the fact that the quantisation does not depend on $\mathbf{B}$. Moreover, as already mentioned at the end of \S~\ref{sec:classical_potentials}, tacking a derivative does not change the type of cycles that make up the potential, but only modifies their weights. This means that the quantisation $\widehat{\partial_{t_j}W_i}$ is well defined, and that taking traces (both of classical and quantum potentials) commutes with picking derivatives.
\end{proof}
Using the second set of identities of the lemma, it is thus enough to verify that one has $\partial_{t_i}\widehat{W_j} - \partial_{t_j}\widehat{W_i} = 0$ for all $i,j \in I$, because the trace of the left-hand side is precisely the difference $\partial_{t_i}\widehat{H}_j - \partial_{t_j}\widehat{H}_i$. Finally, to prove this, one exploits Thm.~\ref{thm:classical_flatness}, borrowing the statement
\begin{equation*}
\partial_{t_i}W_j = \partial_{t_j}W_i.
\end{equation*}
This is precisely Prop.~\ref{prop:direct_verification}, which implies that
\begin{equation*}
\widehat{\partial_{t_i}W_j} = \widehat{\partial_{t_j}W_i}.
\end{equation*}
Then the first set of identity of the above lemma permits to conclude. Notice that crucial fact that the quantisation is ``symmetric on $\mathcal{G}$'', in the sense that the quantisation $\widehat{H}_i$ of the Hamiltonian $H_i$ does not depend on the base node $i \in I$.
\section{Proof of strong flatness: II}
\label{sec:flatness_II}
One is left to show that the quantum IMD Hamiltonians commute. By bilinearity, this reduces to the problem of computing commutators of the form
\begin{equation*}
\big[\Tr(\widehat{C}_1),\Tr(\widehat{C}_2)\big] \in A
\end{equation*}
where $\widehat{C}_1, \widehat{C}_2$ are quantum IMD cycles.
\subsection*{Commutators of quantum cycles}
\ \\
The first thing to do is to see whether one can still write this element in terms of traces of quantum cycles. Let us get back to our quantum algebra $A^{\mathcal{G}}$ whose commutator is defined by $\Tr\big([\widehat{C}_1,\widehat{C}_2]\big) = \big[\Tr(\widehat{C}_1),\Tr(\widehat{C}_2)\big]$.
We would like to be able to give a characterization of $[\widehat{C}_1,\widehat{C}_2]$ along the lines of Prop.~\ref{prop:classicalbrackets}, but unfortunately that used the commutativity of the product on $A_0$. This means that we cannot a priori hope that the commutator of quantum cycles be a quantum potential: one must a priori allow for higher-order elements. \\
However, one can show that the desired property holds for the cycles we're dealing with. Set $\widehat{\IMD} \subseteq \widehat{\comp\mathcal{G}_{\cycl}}$ to be the vector space spanned by the quantum IMD cycles.
\begin{prop}
\label{prop:quantum_commutators}
The restriction $[,]: \widehat{\IMD} \wedge \widehat{\IMD} \longrightarrow A^{\mathcal{G}}$ takes values into $\widehat{\comp\mathcal{G}_{\cycl}}$. Moreover, the commutator $\big[\widehat{C}_1,\widehat{C}_2\big]$ is a quantisation of $\{C_1,C_2\}$, for $\widehat{C}_1,\widehat{C}_2 \in \widehat{\IMD}$ and $C_i := \sigma(\widehat{C}_i)$.
\end{prop}
We will discuss how to control the choice of anchoring for $\big[\widehat{C}_1,\widehat{C}_2\big]$ in the following two sections. For now, let us be content that the commutator between IMD quantum cycles is a linear combination of quantum cycles, instead of a generic polynomial of such. Moreover, the computations for those commutators are basically the same as for the Poisson bracket of classical IMD cycles.
The proof of Prop.~\ref{prop:quantum_commutators} relies on a lemma, plus two separate verifications, whose proofs have been postponed to \S~\ref{sec:appendix}.
\begin{lem}
\label{lem:quantum_commutators}
Pick two quantum cycles $\widehat{C}_1,\widehat{C}_2$, with semiclassical limit $C_1,C_2$. Assume that one of $\widehat{C}_1,\widehat{C}_2$ is a $2$-cycle, or that one of them does not contain pairs of antiparallel arrows. Then Prop.~\ref{prop:quantum_commutators} holds for $\big[\widehat{C}_1,\widehat{C}_2\big] \in \widehat{\comp\mathcal{G}_{\cycl}}$.
\end{lem}
The only IMD cycles that do not satisfy the hypothesis are the degenerate $4$-cycles. Hence one must still show that the commutator of two such cycles follows the same rule. This leads us to check the possible ``intersections'' of cycles in $\mathcal{G}$.
\begin{defn}
Two (classical or quantum) cycles are said to intersect if there exists an arrow of the first with its antiparallel in the second. The intersection is said to be nontrivial if the two cycles are different.
\end{defn}
Notice that two classical cycles (\textit{resp.} quantum cycles) may have a nonvanishing Poisson bracket (\textit{resp.} vanishing commutator) only if they intersect nontrivially. Now, two degenerate $4$-cycles have only two possible nontrivial intersections: either they have the centre in common, or they do not.
\begin{prop}
Pick nodes $a,b,c,d \in I$ such that the sequences of nodes $(a,b,a,c)$ and $(a,c,d,c)$ define two degenerate $4$-cycles. Then the following commutator vanishes:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4);
\vertex (a) at (1,1) [label = below:a, fill = black] {};
\vertex (b) at (3,1) [label = below:b] {};
\vertex (c) at (1,3) [label = above:c] {};
\vertex (d) at (4,1) [label = below:a] {};
\vertex (e) at (4,3) [label = above:c, fill = black] {};
\vertex (f) at (6,3) [label = above:d] {};
\node (g) at (3.5,1) {\textbf{,}};
\draw (6.5,0) -- (7,0) -- (7,4) -- (6.5,4);
\draw (8,1.9) -- (8.5,1.9);
\draw (8,2.1) -- (8.5,2.1);
\draw (9,2) ellipse (5pt and 7pt);
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b)
(a) edge [->, line width = 1.3, bend left = 20] (c)
(a) edge [<-, line width = 1.3, bend right = 20] (c)
(e) edge [->, line width = 1.3, bend left = 20] (d)
(e) edge [<-, line width = 1.3, bend right = 20] (d)
(e) edge [->, line width = 1.3, bend left = 20] (f)
(e) edge [<-, line width = 1.3, bend right = 20] (f);
\end{tikzpicture}
\end{center}
\end{prop}
Here we sketched quantum cycle by drawing a black node where their anchor starts.
The next intersection asks instead to show that the following picture is true:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4) ;
\vertex (a) at (1,1) {};
\vertex (b) at (2,2) [fill = black] {};
\vertex (c) at (2,3.4) {};
\vertex (d) at (4,3.4) {};
\vertex (e) at (4,2) [fill = black] {};
\vertex (f) at (5,1) {};
\draw (5.5,0) -- (6,0) -- (6,4) -- (5.5,4);
\draw (7,1.9) -- (7.5,1.9);
\draw (7,2.1) -- (7.5,2.1);
\vertex (g) at (8,1) [label = left:3] {};
\vertex (h) at (9,2) [fill = black] {};
\vertex (i) at (9,3.4) [label = above:2] {};
\vertex (j) at (10,1) [label = right:1] {};
\draw (11,2) -- (11.5,2);
\vertex (k) at (12,1) [label = left:2] {};
\vertex (l) at (13,2) [fill = black] {};
\vertex (m) at (13,3.4) [label = above:3] {};
\vertex (n) at (14,1) [label = right :1] {};
\node (o) at (3,2) {\textbf{,}};
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b)
(c) edge [->, line width = 1.3, bend left = 20] (b)
(c) edge [<-, line width = 1.3, bend right = 20] (b)
(d) edge [->, line width = 1.3, bend left = 20] (e)
(d) edge [<-, line width = 1.3, bend right = 20] (e)
(f) edge [->, line width = 1.3, bend left = 20] (e)
(f) edge [<-, line width = 1.3, bend right = 20] (e)
(g) edge [->, line width = 1.3, bend left = 20] (h)
(g) edge [<-, line width = 1.3, bend right = 20] (h)
(i) edge [->, line width = 1.3, bend left = 20] (h)
(i) edge [<-, line width = 1.3, bend right = 20] (h)
(j) edge [->, line width = 1.3, bend left = 20] (h)
(j) edge [<-, line width = 1.3, bend right = 20] (h)
(k) edge [->, line width = 1.3, bend left = 20] (l)
(k) edge [<-, line width = 1.3, bend right = 20] (l)
(m) edge [->, line width = 1.3, bend left = 20] (l)
(m) edge [<-, line width = 1.3, bend right = 20] (l)
(n) edge [->, line width = 1.3, bend left = 20] (l)
(n) edge [<-, line width = 1.3, bend right = 20] (l);
\end{tikzpicture}
\end{center}
The number at the peripheral nodes indicates the order in which one must touch them, starting from the centre (the tail of the anchor).
\subsection*{Anchors}
\ \\
Let us decompose the classical IMD potentials $W_i,W_j$ into a sum of classical IMD cycles: $W_i = \sum_k c_k C_k, W_j = \sum_l d_l D_l$. After expanding their vanishing Poisson bracket by bilinearity, one will find itself with a sum of potentials:
\begin{equation*}
0 = \{W_i,W_j\} = \sum_{k,l} c_kl_l \{C_k,D_l\}.
\end{equation*}
Putting together all the cycles that coincide as elements of $\comp\mathcal{G}_{\cycl}$, one will get to a finer decomposition
\begin{equation*}
0 = \{W_i,W_j\} = \sum_m e_m E_m \in \comp\mathcal{G}_{\cycl}.
\end{equation*}
Now, since we're assuming that $E_m \neq E_{m'}$ for $m \neq m'$ in this sum, one has necessarily $e_m = 0$ for all $m$: any finite family of distinct cycles in $\mathcal{G}$ is free inside $\comp\mathcal{G}_{\cycl}$, by definition.
Now, thanks to Prop.~\ref{prop:quantum_commutators}, one will find a similar development:
\begin{equation*}
\big[\widehat{W}_i,\widehat{W}_j\big] = \sum_{k,l} c_{kl} \big[\widehat{C}_k,\widehat{D}_l\big],
\end{equation*}
with $\widehat{C}_k, \widehat{D}$ being the quantisation of $C_k, D_l$. Moreover, $\big[\widehat{C}_k,\widehat{D}_l\big]$ is a quantisation of $\{C_k,D_l\}$. One would now hope to have
\begin{equation*}
\big[\widehat{W}_i,\widehat{W}_j\big] = \sum_m e_m \widehat{E}_m \in \widehat{\comp\mathcal{G}_{\cycl}},
\end{equation*}
with the same constants $e_m \in \comp$, for some lift $\widehat{E}_m$ of $E_m$. This happens if and only if every time that one has $\{C_k,D_l\} = \{C_{k'},D_{l'}\}$ in $\comp\mathcal{G}_{\cycl}$, then one also has $\big[\widehat{C}_k,\widehat{D}_l\big] = \big[\widehat{C}_{k'},\widehat{D}_{l'}\big]$ in $\widehat{\comp\mathcal{G}_{\cycl}}$. Since those two commutators have the same underlying classical cycle, this happens if and only if their anchors are equivalent. The obstruction for this to happen is that above a given intersection of classical IMD cycles there are several nonequivalent intersections of quantum IMD cycles: the anchor breaks some symmetry, \textit{a priori}. The final part of the proof of Thm. \ref{thm:quantum_flatness} consists in showing that this does not happen.
First, a direct verification based on elementary combinatoric arguments shows the following.
\begin{prop}
There exist exactly 15 distinct nontrivial intersections of classical IMD cycles. Among them, 13 give a nonvanishing Poisson bracket, and 5 of those are sums of cycles without pairs of antiparallel arrows.
\end{prop}
To prove this proposition, the position of the anchor of all $2$-cycles is immaterial. Indeed, thanks to Prop.~\ref{prop:change_anchor}, moving the anchor amounts to add a constant, which lies in the centre of $A$. Now, all the cases where one has no pairs of antiparallel arrows give no issues: any two quantisations of a cycle without such pairs are equivalent. One must thus consider the remaining 8 ``troublesome'' intersections, and see that no symmetry can be broken by adding an anchor to the cycles involved. Those intersections can be described as follows, in plain words:
\begin{enumerate}
\item two opposite $3$-cycles
\label{one}
\item a $3$-cycle and a degenerate $4$-cycle with one pair of antiparallel arrows in common
\label{two}
\item a $3$-cycle and a nondegenerate $4$-cycle with two pairs of antiparallel arrows in common
\label{three}
\item two nondegenerate $4$-cycles with the centre in common
\label{four}
\item a nondegenerate $4$-cycle and a degenerate one, with one pair of antiparallel arrows in common
\label{five}
\item same as the one just above, with two pairs in common
\label{six}
\item two nondegenerate $4$-cycles with two pairs of antiparallel arrows in common
\label{seven}
\item two opposite nondegenerate $4$-cycles
\label{eight}
\end{enumerate}
One can finally discuss those separately. All these intersections give cycles which are classically distinguishable, apart from the pairs \big(\ref{two},\ref{three}\big) and \big(\ref{five},\ref{seven}\big). Also, n\textdegree \ref{four} has already been dealt with above.
\subsection{Last verifications}
\label{sec:last_verifications}
\ \\
Here we argue that the aforementioned nontrivial intersections \ref{one} -- \ref{eight} yield equivalent quantum potentials, as needed in order to conclude the proof of Thm. \ref{thm:quantum_flatness}. We will thus sketch a few commutators of quantum cycles. The logic behind the pictures is always to summarise longer computations in (noncommutative) variables, exploiting Prop.~\ref{prop:quantum_commutators}. Notice however that the explicit computations appear in the appendix \S~\ref{sec:appendix}, where a harmless choice of Darboux coordinates is made in order to simplify the constants that come out of the commutators.
First, n\textdegree\ref{one} and n\textdegree\ref{eight} are settled by a uniqueness argument: in both cases, two such pairs appears exactly twice in the commutator $\big[\widehat{W}_i,\widehat{W}_j\big]$, and with reversed orders. This just gives a sign, and the vanishing of the associated weights follows from Thm. \ref{thm:classical_flatness}: if this did not happen, then the classical IMD system would not be flat.
Next, let us move to n\textdegree\ref{two} and n\textdegree\ref{three}. One can verify that those nontrivial intersections produce $5$-cycles built from glueing a $2$-cycle to a $3$-cycle, the two having no antiparallel arrows in common. It would then be enough to choose anchors so that one always follows the $3$-cycle first, and this can indeed be done.
\begin{prop}
Pick nodes $a,b,c,d \in I$ so that $(a,d,c)$ defines a $3$-cycle. Assume also that $a$ and $b$ are adjacent.
Then one may choose Darboux coordinates so that:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4);
\vertex (a) at (1,1) [label = below:a, fill = black] {};
\vertex (b) at (1,3) [label = above:b] {};
\vertex (c) at (3,1) [label = below:c] {};
\vertex (d) at (4,1) [label = below:a, fill = black] {};
\vertex (e) at (6,3) [label = above:d] {};
\vertex (f) at (6,1) [label = below:c] {};
\draw (6.5,0) -- (7,0) -- (7,4) -- (6.5,4);
\draw (8,1.9) -- (8.4,1.9);
\draw (8,2.1) -- (8.4,2.1);
\vertex (g) at (9,1) [label = below:a, fill = black] {};
\vertex (h) at (9,3) [label = above:b] {};
\vertex (i) at (11,1) [label = below:c] {};
\vertex (j) at (11,3) [label = above:d] {};
\node (k) at (3.5,1) {\textbf{,}};
\path
(a) edge [->, line width = 1.3, bend left = 20] (b)
(a) edge [<-, line width = 1.3, bend right = 20] (b)
(a) edge [->, line width = 1.3, bend left = 20] (c)
(a) edge [<-, line width = 1.3, bend right = 20] (c)
(d) edge [->, line width = 1.3] (e)
(e) edge [->, line width = 1.3] (f)
(f) edge [->, line width = 1.3] (d)
(g) edge [->, line width = 1.3, bend left = 20] (h)
(g) edge [<-, line width = 1.3, bend right = 20] (h)
(g) edge [->, line width = 1.3] (j)
(j) edge [->, line width = 1.3] (i)
(i) edge [->, line width = 1.3] (g);
\end{tikzpicture}
\end{center}
\end{prop}
\begin{prop}
Pick nodes $a,b,c,d \in I$ defining a $4$-cycle. Assume that $a$ and $c$ are adjacent. Then one may choose Darboux coordinates so that:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4);
\vertex (a) at (1,1) [label = below:a, fill = black] {};
\vertex (b) at (3,1) [label = below:b] {};
\vertex (c) at (3,3) [label = above:c] {};
\vertex (d) at (1,3) [label = above:d] {};
\vertex (e) at (4,1) [label = below:a, fill = black] {};
\vertex (f) at (6,3) [label = above:c] {};
\vertex (g) at (6,1) [label = below:b] {};
\draw (6.5,0) -- (7,0) -- (7,4) -- (6.5,4);
\draw (8,1.9) -- (8.4,1.9);
\draw (8,2.1) -- (8.4,2.1);
\vertex (h) at (9,1) [label = below:a, fill = black] {};
\vertex (i) at (11,3) [label = above:c] {};
\vertex (j) at (9,3) [label = above:d] {};
\vertex (k) at (11,1) [label = below:b] {};
\draw (11.6,2) -- (12,2);
\draw (11.8,1.8) -- (11.8,2.2);
\vertex (l) at (12.5,1) [label = below:a, fill = black] {};
\vertex (m) at (14.5,3) [label = above:c] {};
\vertex (n) at (12.5,3) [label = above:d] {};
\vertex (o) at (14.5,1) [label = below:b] {};
\node (p) at (3.5,1) {\textbf{,}};
\path
(a) edge [->, line width = 1.3] (b)
(b) edge [->, line width = 1.3] (c)
(c) edge [->, line width = 1.3] (d)
(d) edge [->, line width = 1.3] (a)
(e) edge [->, line width = 1.3] (f)
(f) edge [->, line width = 1.3] (g)
(g) edge [->, line width = 1.3] (e)
(h) edge [->, line width = 1.3] (i)
(i) edge [->, line width = 1.3] (j)
(j) edge [->, line width = 1.3] (h)
(i) edge [->, line width = 1.3, bend left = 20] (k)
(i) edge [<-, line width = 1.3, bend right = 20] (k)
(l) edge [->, line width = 1.3] (m)
(l) edge [->, line width = 1.3, bend left = 20] (o)
(l) edge [<-, line width = 1.3, bend right = 20] (o)
(m) edge [->, line width = 1.3] (n)
(n) edge [->, line width = 1.3] (l);
\end{tikzpicture}
\end{center}
\end{prop}
Now, every time that such a nontrivial intersection arises, one can base the $3$-cycle as in the above figure without loss of generality, and the resulting $5$-cycle will start at its $3$-subcycle. In particular, two such commutators will equal if and only if their associated classical brackets are, which is the result one is after.
Next, one should consider n\textdegree\ref{six}.
\begin{prop}
Pick nodes $a,b,c,d \in I$ defining a $4$-cycle. One may choose Darboux coordinates so that:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4);
\vertex (a) at (1,1) [label = below:a, fill = black] {};
\vertex (b) at (3,1) [label = below:b] {};
\vertex (c) at (1,3) [label = above:d] {};
\vertex (d) at (3,3) [label = above:c] {};
\vertex (e) at (4,1) [label = below:a] {};
\vertex (f) at (6,1) [label = below:b, fill = black] {};
\vertex (g) at (6,3) [label = above:c] {};
\draw (6.5,0) -- (7,0) -- (7,4) -- (6.5,4);
\draw (7.7,1.9) -- (8.2,1.9);
\draw (7.7,2.1) -- (8.2,2.1);
\vertex (h) at (9,1) [label = below:a] {};
\vertex (i) at (11,1) [label = below:b, fill = black] {};
\vertex (j) at (9,3) [label = above:d] {};
\vertex (k) at (11,3) [label = above:$c_2$] {};
\vertex (p) at (12,3) [label = above:$c_1$] {};
\draw (12.6,2) -- (12.9,2);
\draw (11.3,2.9) -- (11.7,2.9);
\draw (11.3,3.1) -- (11.7,3.1);
\vertex (l) at (13.5,1) [label = below:a, fill = black] {};
\vertex (m) at (15.5,1) [label = right:$b_2$] {};
\vertex (n) at (13.5,3) [label = above:d] {};
\vertex (o) at (15.5,3) [label = above:c] {};
\vertex (q) at (15.5,0) [label = right:$b_1$] {};
\node (r) at (3.5,1) {\textbf{,}};
\draw (15.4,0.7) -- (15.4,0.3);
\draw (15.6,0.7) -- (15.6,0.3);
\path
(a) edge [->, line width = 1.3] (b)
(b) edge [->, line width = 1.3] (d)
(d) edge [->, line width = 1.3] (c)
(c) edge [->, line width = 1.3] (a)
(e) edge [->, line width = 1.3, bend left = 20] (f)
(e) edge [<-, line width = 1.3, bend right = 20] (f)
(f) edge [->, line width = 1.3, bend left = 20] (g)
(f) edge [<-, line width = 1.3, bend right = 20] (g)
(h) edge [->, line width = 1.3] (i)
(i) edge [->, line width = 1.3] (k)
(k) edge [->, line width = 1.3] (j)
(j) edge [->, line width = 1.3] (h)
(l) edge [->, line width = 1.3] (m)
(m) edge [->, line width = 1.3] (o)
(o) edge [->, line width = 1.3] (n)
(n) edge [->, line width = 1.3] (l)
(i) edge [->, line width = 1.3, bend left = 20] (p)
(i) edge [<-, line width = 1.3, bend right = 20] (p)
(l) edge [->, line width = 1.3, bend left = 20] (q)
(l) edge [<-, line width = 1.3, bend right = 20] (q);
\end{tikzpicture}
\end{center}
\end{prop}
On the right-hand side one has split the nodes $c = c_1 = c_2$ and $b = b_1 = b_2$, so to indicate the order in which they're touched. The point of this proposition is the same as before: up to changing the anchor of the nondegenerate $4$-cycle, all $6$-cycles that appear as a result of this type of nontrivial intersection will have equivalent anchors (one follows the $2$-cycle first).
Finally, one should check n\textdegree\ref{five} and n\textdegree\ref{seven}. Those two nontrivial intersections produce $6$-cycles built from glueing a nondegenerate $4$-cycle and a $2$-cycle, the two having no antiparallel arrows in common. It would then be enough to choose anchors so that one always follows the $4$-cycle first, and this can indeed be done.
\begin{prop}
Pick nodes $a,b,c,d,e \in I$ such that $(a,b,c,d)$ defines a $4$-cycle. Assume that $b$ and $e$ are adjacent. Then one can choose Darboux coordinates so that:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4);
\vertex (a) at (1,1) [label = below:a, fill = black] {};
\vertex (b) at (3,1) [label = below:b] {};
\vertex (c) at (1,3) [label = above:d] {};
\vertex (d) at (3,3) [label = above:c] {};
\vertex (e) at (4,1) [label = below:a] {};
\vertex (f) at (6,1) [label = below:b, fill = black] {};
\vertex (g) at (5,2.4) [label = above:e] {};
\draw (6.5,0) -- (7,0) -- (7,4) -- (6.5,4);
\draw (8,1.9) -- (8.5,1.9);
\draw (8,2.1) -- (8.5,2.1);
\vertex (h) at (9,1) [label = below:a, fill = black] {};
\vertex (i) at (11,1) [label = below:b] {};
\vertex (j) at (9,3) [label = above:d] {};
\vertex (k) at (11,3) [label = above:c] {};
\vertex (l) at (10,2.4) [label = below left:e] {};
\node (m) at (3.5,1) {\textbf{,}};
\path
(a) edge [->, line width = 1.3] (b)
(b) edge [->, line width = 1.3] (d)
(d) edge [->, line width = 1.3] (c)
(c) edge [->, line width = 1.3] (a)
(e) edge [->, line width = 1.3, bend left = 20] (f)
(e) edge [<-, line width = 1.3, bend right = 20] (f)
(f) edge [->, line width = 1.3, bend left = 20] (g)
(f) edge [<-, line width = 1.3, bend right = 20] (g)
(h) edge [->, line width = 1.3] (i)
(i) edge [->, line width = 1.3] (k)
(k) edge [->, line width = 1.3] (j)
(j) edge [->, line width = 1.3] (h)
(i) edge [->, line width = 1.3, bend left = 20] (l)
(i) edge [<-, line width = 1.3, bend right = 20] (l);
\end{tikzpicture}
\end{center}
\end{prop}
\begin{prop}
Pick nodes $a,b,c,d,e \in I$ so that $(a,b,c,d)$ and $(a,b,c,e)$ define $4$-cycles. One can choose Darboux coordinates so that:
\begin{center}
\begin{tikzpicture}
\draw (0.5,0) -- (0,0) -- (0,4) -- (0.5,4);
\vertex (a) at (1,1) [label = below:a, fill = black] {};
\vertex (b) at (3,1) [label = below:b] {};
\vertex (c) at (3,3) [label = above:c] {};
\vertex (d) at (1,3) [label = above:d] {};
\vertex (e) at (4,1) [label = below:a] {};
\vertex (f) at (5,2) [label = above:e] {};
\vertex (g) at (6,3) [label = above:c] {};
\vertex (h) at (6,1) [label = below:b, fill = black] {};
\draw (6.5,0) -- (7,0) -- (7,4) -- (6.5,4);
\draw (8,1.9) -- (8.5,1.9);
\draw (8,2.1) -- (8.5,2.1);
\vertex (i) at (9,1) [label = below:a, fill = black] {};
\vertex (j) at (10,2) [label = below:e] {};
\vertex (k) at (11,3) [label = above:c] {};
\vertex (l) at (11,1) [label = below:b] {};
\vertex (m) at (9,3) [label = above:d] {};
\draw (11.6,2) -- (11.9,2);
\vertex (n) at (12.5,1) [label = below:a, fill = black] {};
\vertex (o) at (13.5,2) [label = right:e] {};
\vertex (p) at (14.5,3) [label = above:c] {};
\vertex (q) at (12.5,3) [label = above:d] {};
\vertex (r) at (14.5,1) [label = below:b] {};
\node (s) at (3.5,1) {\textbf{,}};
\path
(a) edge [->, line width = 1.3] (b)
(b) edge [->, line width = 1.3] (c)
(c) edge [->, line width = 1.3] (d)
(d) edge [->, line width = 1.3] (a)
(e) edge [->, line width = 1.3] (f)
(f) edge [->, line width = 1.3] (g)
(g) edge [->, line width = 1.3] (h)
(h) edge [->, line width = 1.3] (e)
(i) edge [->, line width = 1.3] (j)
(j) edge [->, line width = 1.3] (k)
(k) edge [->, line width = 1.3, bend left = 20] (l)
(k) edge [<-, line width = 1.3, bend right = 20] (l)
(k) edge [->, line width = 1.3] (m)
(m) edge [->, line width = 1.3] (i)
(n) edge [->, line width = 1.3] (o)
(o) edge [->, line width = 1.3] (p)
(p) edge [->, line width = 1.3] (q)
(q) edge [->, line width = 1.3] (n)
(n) edge [->, line width = 1.3, bend left = 20] (r)
(n) edge [<-, line width = 1.3, bend right = 20] (r);
\end{tikzpicture}
\end{center}
\end{prop}
This concludes the proof of Thm. \ref{thm:quantum_flatness}.
| 187,321
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More Audience Reviews
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AUDIENCE REVIEW: Arim Dance in "Currents"
Company:
Arim Dance
Performance Date:
November 10 & 11, 2018
Have you ever seen this company/ before?
Tell us a bit about your history with this group/performer?
I saw Arim Dance perfrom at the Alvin Ailey Theatre. I was there to view the up and coming company which is run by Martha Graham Dance Ari Mayzick.
What was your favorite moment(s)? What inspired you?
I saw the performance of Currents and was astounded by the stamina and technique required to perform the piece. The choreography was extremely complex and created a feeling of vigurous ocean currents.
Describe as plainly and specifically as you can what actually happened during this performance.
Dancers Sarita Apel, Sarita Apel, Carley Marholin, Whitney Janis, Samantha McLoughlin, Jennifer Nelson, Esteban Santamaria, Thibaut Witzleb, Sarah Grace Houston and Sophy Zhao worked wonderfully as one unit. It seems like the dance was made to have no dancer stand but to have the group shine as one, which they did.
Do any images, colors or feelings pop into your head when you think about this show?
I remember seeing the colors of the sea, greens, blues and indigo.
Describe any or all of these elements: music, lighting, the venue.
How did they contribute (or not) to your enjoyment of this performance?
The music was wonderful, very classical and was neat to see classical music with Martha Graham technique. Everything worked together nicely.
Would you like to see this performance / company again?
I would love to see them again.
| 101,258
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Pushing through, after a very draining day. But here's #11 of 30 in my November Poetry Challenge of 30 poems in 30 days. The next few poems will be posted as a lump next week, because I'm going to be out of town this weekend. (Yes, there will still be folks here at the house. I'm not careless enough to announce on line when the home will be empty.) I'll be out at , and will likely have a most wonderful time. But tonight...
Creeping Inertia
I'm supposed to be on site tomorrow,
raring to go,
camp established,
deep into my role of Mistress Gwen,
lost in the sparkling depths
of a shared time out of time.
And here I sit.
No packing list made.
No projects chosen.
No food prepped.
Not even any laundry done.
And no motivation at all.
I know I'll have fun once I get there,
but right now the insidious
Inertia Monster
has me dangling limp in his claws,
and I've got a bad case
of been there,
done that.
Maybe I can shake him
in my sleep.
Wish me luck...
Melissa McCollum
11/11/2010
| 96,527
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Dragon Age: Origins Ultimate Edition detailed
BBFC has rated the Ultimate Edition of Dragon Age: Origins giving us the hint that the Bioware's RPG will be re-releasing soon. However apart from rating no other details (price or release date) was reveal.
According to BBFC Dragon Age: Origin Ultimate Edition will consist of all the DLC of game, which means all the below DLC will be included as part of Ultimate Edition:
- The Stone Prisoner
- Wardens Keep, Return to Ostagar
- Awakening
- Feast Day Combo Pack
- The Darkspawn Chronicles
- Leliana's Song
- The Golems of Amgarrak and Witch Hunt
Dragon is slated to release on March 8.
| 153,585
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We are always interested in finding creative ways to communicate with people about migration. We can provide comments, conduct interviews and provide Wales wide statistics on specific matters of migration that can inform and engage with a wide range of media audiences
If you have an idea or want to develop a story regarding any issues please get in touch.
Film Projects
The Welsh Refugee Council has worked with the Welsh Government and DPiA to develop a short film on Hate Crime. The project focuses on the experiences of three asylum seekers and refugees and their recommendations for future action. Viewed by over 300 people at the launch of the Welsh Government’s Hate Crime Framework, the film starkly illustrates the need for action with regard to this issue.
Photography projects
The Welsh Refugee Council is currently working with photographer Marcus Oleniuk on an Arts Council funded project entitled ‘Stryd Loches’ or ‘Asylum Street’. The project is collaborative and involves photography workshops for refugees and asylum seekers. Following these sessions, the participants will be provided with cameras and can begin documenting their asylum experience in Wales.
The ‘Stryd Loches’ project provides vulnerable people with a creative outlet to unlock and express internalised thoughts and memories. This self-expression helps individuals confront and transcend the trauma of fleeing persecution, whilst making others aware of the asylum experience in Wales.
The principal aims include:
| 164,310
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TITLE: Question Regarding The Ideal Gas Law
QUESTION [2 upvotes]: I have been studying thermodynamics and heat transfers and there I met with the ideal gas law, in fact I have used it a lot of times until now, but when i see the proof of this law, I saw that it has been derived by combining Boyle's law, Charles' law, and the third one is Avogadro's law I think, so now i am confused because Boyle's law assumes that Temperature of the gas is constant, Charles' law assumes that the Pressure of the gas is constant and Avogadro's law says that temperature and pressure both must be constant, so how can we combine all these proportional pieces into one single cake, when all these pieces have different dependencies ?
REPLY [1 votes]: Phillip Wood provided a very useful summary, mentioning various necessary ideas. However, I recall myself as a student of thermodynamics having similar questions, and these answers still never completely satisfied me and I perceive that it still might not completely answer the concern. Really, the question already states that the various laws have already been outlined and brought together in the IDE, so I'm not convinced that even the best summary like Phillip's really answer the question.
The steps of a scientific method are taught in school, but that method is hardly ever revealed in explaining how many scientific laws were obtained. I think an understanding of the historical process would be enlightening and fill in gaps. Regrettably I'm not the historian to share the actual details. But it would show a process in which the Ideal Gas Law could not be postulated first, one in which the concept of especially temperature was still being explored and not fully understood, one in which the idea of separate particles (i.e. atoms and particles) was not yet taken for granted... and was sometimes taken as bogus. We'd see how various experimental results and incomplete formulas were scratched on various sheets and in notebooks. Results would be shared in letters and publish in journals, but at far slower speed than we realize today. Technology had to improve along with the science, otherwise there was no good way to measure and/or maintain constant temperatures... or to keep a gas sealed without leaking out of suitable containers, etc.
Regardless of what intuitive ideas we have developed today, some of what we consider simple relationships could not be taken for granted in the past. Using rather basic measurement devices, the scientists would necessarily choose to keep one or many things constant. It would have been nigh impossible to do experiments and learn anything if all aspects were wildly fluctuating. And then it took a long time in sharing results and for others to piece all the experimental results together, probably with a lot of incorrect or incomplete formulas along the way.
So if one excludes the possibility of postulating the ideal gas, the answer to
... how can we combine all these proportional pieces into one single cake, when all these pieces have different dependencies?
is that first of all it is absolutely necessary to perform separate experiments, keeping at least one of the dependencies constant in each. Then one-by-one recognize similarities, do some mathematical manipulations and relate constants, make some guesses (i.e. hypotheses)--including wrong ones, then finally deduce the correct relationship and re-test.
Pardon me if this sounds like a stupid answer, but I recall that nearly all thermodynamic explanations always started with some sort of postulate / axiom that was plucked from air and nobody (in my realm) really elucidated how or where such axioms came from. I've realized it is because there is no neat outline or short answer, none that anyone could explain easily and so they didn't really try.
| 139,927
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After making yogurt and experimenting with some basic flavorings, I've decided to take more of the advice of my yogurt-making instructions to flavor it in new and exciting ways..
What have I learned so far? Combination sweetener-flavors are more effective than either alone; the more liquid an additive is, the better texture the yogurt ends up with; and if anything, lean toward more flavoring/sweetening than is suggested.
What's on board next? Smaller batches of more "out-there" flavors (any suggestions?). And, eventually, frozen yogurt!
Wednesday, March 21, 2012
Yogurt Chronicle, Part III
Posted by Matt Pavlovich at 4:35 PM 3 comments:
Labels: Food, Yogurt Chronicle
| 130,632
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It's time to begin laying plans for programs to celebrate Women's Equality
Day for your workplace, community, or organizations. It can be far easier
than you may think!
Everything you need for a successful Women's Equality Day program is
available by mail from the National Women's History Project: posters and
display sets, short videos and well-crafted speeches, colorful placemats,
imprinted balloons and buttons.
Ask for a free "Women's History Catalog" and join the national celebration!
Contact the National Women's History Project at <<nwhp@aol.com>>
August 26 has been observed since 1971, when the U.S. Congress designated the
day to honor women's continuing efforts toward equality. The date is
historically significant: It was on August 26, 1920, that the 19th Amendment
to the U.S. Constitution was finally ratified, granting women the right to
vote after their 72-year campaign.
| 255,714
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Storytime at Historic Railpark &Train Museum
The Historic RailPark & Train Museum will offer Storytime at the RailPark during the month of July every Tuesday and Saturday morning at 10:00am.
The community is invited to join us for an hour of interactive stories, craft time and an accompanying activity perfect for young children. Supplies for crafts will be provided by the RailPark. Activities will include a variety of sensory bins and physical movement.
For parents and caregivers, Storytime offers an opportunity to experience early literacy practices in action, to discover great books and socializing with other parents after Storytime. Each week we offer a different book along with coordinating activities.
Storytime at the RailPark will be held during July on Tuesdays and Saturdays at 10:00 am at the L&N Depot located in Downtown Bowling Green.
Story Time at the RailPark July Book Schedule is as follows:
Week 1 July 5th & 9th – Thomas Comes to Breakfast
Week 2 July 12th & 16th – Chuggington Lights, Camera, Action Chugger!
Week 3 July 19th & 23rd – I Saw an Ant on the Railroad Track
Week 4 July 26th & 30th – The Caboose Who Got Loose
If you have questions about Storytime at the RailPark please visit us at 401 Kentucky Street in Downtown Bowling Green or by call 270-745-7317.
For more information call 270-745-7317 or visit historicrailpark.com
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\begin{document}
\title{Symbolic extensions for 3-dimensional diffeomorphisms}
\author[David Burguet and Gang Liao]{David Burguet $^{*}$ and Gang Liao $^{**}$}
\email{david.burguet@upmc.fr}
\email{lg@suda.edu.cn}
\date{November, 2019}
\keywords{Symbolic extension; 3-dimensional diffeomorphism; tail entropy}
\thanks{2010 {\it Mathematics Subject Classification}. 37B10, 37A35, 37C40}
\thanks{$^*$LPSM, Sorbonne Universite, Paris 75005, France. $^{**}$School of Mathematical Sciences, Center for Dynamical Systems and Differential Equations, Soochow University,
Suzhou 215006, China; G. Liao was partially supported by NSFC (11701402, 11790274), BK 20170327 and IEPJ}
\maketitle
\begin{abstract}We prove that every $\mathcal{C}^{r}$ diffeomorphism with $r>1$ on a three-dimensional manifold admits symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. This answers positively a conjecture of Downarowicz and Newhouse in dimension three.
\end{abstract}
\allowdisplaybreaks
\section{Introduction}\label{sec:intro}
A symbolic extension of a topological dynamical system is a topological extension given by a subshift over a finite alphabet. Existence and entropy of symbolic extensions have been intensively investigated in the last decades. M. Boyle and T. Downarowicz \cite{BD} characterized the problem of existence in terms of new entropic invariants related to weak expansiveness properties of the system. In particular asymptotically $h$-expansive systems always admit \textit{principal} symbolic extensions, i.e. extensions that preserve the entropy of invariant measures \cite{bff}.
For smooth systems on compact manifolds this theory appears to be of highly interest. It is well known that Markov partitions allow to encode uniformly hyperbolic systems by finite-to-one symbolic extensions of finite type. Beyond uniform hyperbolicity, partially hyperbolic diffeomorphisms with one dimensional center satisfy the $h$-expansiveness property, hence admit principal symbolic extensions \cite{CY, LFPV}. More recently the second author with M. Viana and J. Yang showed that smooth systems with no principal symbolic extension are $\mathcal{C}^1$-close to diffeomorphisms with homoclinic tangencies \cite{LVY}.
Moreover the existence of symbolic extensions depends on the order of smoothness. While $\mathcal{C}^\infty$ systems are asymptotically $h$-expansive \cite{Buz, Yom} and thus admit principal symbolic extensions, there is a $\mathcal{C}^1$ open set of 3-dimensional diffeomorphisms \cite{asa} (resp. Lebesgue preserving diffeomorphisms \cite{BCF, DN}) in which generic ones have no symbolic extension.
In intermediate smoothness, i.e. for $\mathcal{C}^r$ systems with $1<r<+\infty$, the existence was conjectured by T. Downarowicz and S. Newhouse in \cite{DN} and in general this problem is still open. It has been first proved for circle maps by T. Downarowicz and A. Maass \cite{DM} and then by the first author for surface diffeomorphisms \cite{Burguet11,Burguet12}. In this paper we work further on \cite{Burguet12} to show existence of symbolic extensions for diffeomorphisms in dimension 3. We refer to the next section for the definitions and notations used in our main Theorem below.
\begin{Mtheorem}\label{maintheorem}
Let $f$ be a $\mathcal{C}^{r}$ diffeomorphism with $r>1$ on a compact 3-dimensional manifold $M$. Then $f$ admits a symbolic extension $\pi:(Y,S)\rightarrow (M,f)$ satisfying for all $\mu\in \mathcal{M}_{inv}(f)$:
$$ \max_{\xi\in \mathcal{M}_{inv}(S), \ \pi \xi=\mu}h(S,\xi)=h(f,\mu)+\frac{\lambda_1^+(f,\mu)+\lambda_2^+(f,\mu)}{r-1},$$
\noindent where $\lambda_1^+(f,\mu)\geq \lambda_2^+(f,\mu)$ denote the positive parts of the two largest Lyapunov exponents of $\mu$.
\end{Mtheorem}
The ingredient of the present advance is mainly a new inequality relating the Newhouse local entropy of an ergodic measure and the local volume growth of smooth discs of unstable dimension (which is the number of positive Lyapunov exponents of the measure).
Section 3 is devoted to the proof of this key estimate. Then for a $3$-dimensional diffeomorphism, we may bound from above the Newhouse local entropy with respect to either $f$ or $f^{-1}$ by the local volume growth of curves, which implies the existence of symbolic extensions by combining with the \textit{Reparametrization Lemma} developed in \cite{Burguet12}. This is proved together with the Main Theorem in the last section.
\section{Preliminaries}\label{sec:pre}
\subsection{ Newhouse entropy structure and the Symbolic Extension Theorem.}
Consider a topological system $(M,f)$, i.e. a continuous map $f:M\rightarrow M$ on a compact metric space $(M,d)$. For $x\in M, \vep>0, n\in \mathbb{N}$, we denote the $n$-step dynamical ball at $x$ with radius $\varepsilon$ by $$B_n(x,\vep,f)=\{y\in M: d(f^i(x), f^i(y))<\vep,\,\,i=0,\cdots,n-1\}.$$ A subset $N$ of $M$ is said $(n,\delta)$-separated when any pair $y\neq z$ in $N$ satisfies $d(f^i(y),f^i(z))>\delta$ for some $i\in [0,n-1]$. For any subset $\Lambda$ of $M$ and $\delta>0$, denote by $s(n,\delta, \Lambda)$ the maximal cardinality of the $(n,\delta)$-separated sets contained in $\Lambda$. For any $\Lambda\subset M$, $\vep>0$, define $$h^*(f,\Lambda, \vep)=\lim_{\delta\to 0}\limsup_{n\to \infty}\frac1n\log \sup_{x\in \Lambda}s\left(n,\delta,B_{n}(x,\vep,f)\cap\Lambda\right).$$
Denote by $\mathcal{M}_{inv}(f)$ (resp. $\mathcal{M}_{erg}(f))$ the set of all $f$-invariant (resp. ergodic $f$-invariant) Borel probability measures endowed with the usual metrizable weak-$*$ topology. Given $\mu\in \mathcal{M}_{erg}(f)$, for any $\vep>0$, Newhouse \cite{New} defined the tail entropy of $\mu$ at the scale $\vep$ by letting
$$h^*(f, \mu, \vep)=\lim_{\eta\to 1,\,0<\eta<1} \inf_{\mu(\Lambda)>\eta} h^*(f,\Lambda, \vep).$$
For $\mu\in \mathcal{M}_{inv}(f)$, assuming
$\mu=\int_{\mathcal{M}_{erg}(f)}\nu \, dM_\mu(\nu)$ is the ergodic
decomposition of $\mu$, let $$h^*(f, \mu, \vep)=\int_{\mathcal{M}_{erg}(f)} h^*(f,\nu, \vep)\,dM_\mu(\nu).$$
Entropy structures are \textit{particular} non-increasing sequences of nonnegative functions defined on $\mathcal{M}_{inv}(f)$ which are converging pointwisely to the Kolmogorov-Sinai entropy function $h:\mathcal{M}_{inv}(f)\rightarrow \mathbb{R}^+$ (see \cite{Dow} for a precise definition). They satisfy the following criterion for the existence of symbolic extensions.
\begin{theorem}[Symbolic Extension Theorem \cite{BD, DM}]\label{SEX}
Let $(M,f)$ be a topological system. Assume $E$ is a nonnegative affine upper semicontinuous function such that for all $\mu\in \mathcal{M}_{inv}(f)$ there is an entropy structure $(h_k)_k$ satisfying
\begin{equation}\label{fon}\lim_k \limsup_{\mathcal{M}_{erg}(f)\ni\nu\rightarrow \mu}(E+h-h_k)(\nu)\leq E(\mu).\end{equation}
Then there exists a symbolic extension $\pi:(Y,S)\rightarrow (M,f)$ such that
$$ \max_{\xi \in \mathcal{M}_{inv}(S), \ \pi \xi=\mu}h(S,\xi)=(E+h)(\mu).$$
\end{theorem}
Letting $\vep_k\to 0$, then the sequence $(h^{New}_k)_k$ defined by $h^{New}_k(f, \mu):=h(f, \mu)-h^*(f, \mu, \vep_k)$ for all $k\in \mathbb{N}$ and for all $\mu\in \mathcal{M}_{inv}(f)$ is an entropy structure \cite{Dow}. As a matter of fact, for any $m\in \mathbb{Z}\setminus\{0\}$, $h^{New}_{m, k}(f, \mu):=h(f, \mu)-\frac{1}{|m|}h^*(f^m, \mu, \vep_k)$ for all $k\in \mathbb{N}$ and for all $\mu\in \mathcal{M}_{inv}(f)$ is also an entropy structure (see Lemma 1 in \cite{Burguet12}).
\subsection{Lyapunov exponents}
Let $f:M\rightarrow M$ be a differentiable map on a compact Riemannian manifold $(M,\|\cdot\|)$ of dimension $\mathsf d$.
Given $x\in M$, the Lyapunov exponent relative to a direction $v\in T_xM$ is the exponential growth rate given by the limit
\begin{eqnarray}\label{limit}\lim_{n\to \infty}\frac{1}{n}\log \|D_xf^nv\|,\end{eqnarray}
which exists for almost every point $x$ with respect to every $f$-invariant Borel probability measure $\mu$ by Oseledets theorem \cite{Oseledets} (it does not depend on the Riemannian structure on $M$). Moreover, for $\mu$-almost every point $x$, there exist values $\lambda_1(f,x)\ge \cdots\ge\lambda_{\mathsf d}(f,x)$ of the limit (\ref{limit}) and measurable flags of the tangent spaces $\{0\}=G^{\mathsf d+1}_x\subset G^{\mathsf d}_x\subset \cdots \subset G^{1}_x=T_xM$ satisfying:
$$\lim_{n\to \infty}\frac{1}{n}\log \|D_xf^nv\|=\lambda_i(f,x),\quad \forall v\in G^i_x\setminus G^{i+1}_x,\,\,1\le i\le \mathsf d. $$
For any $\mu\in \mathcal{M}_{inv}(f) $, $1\le i\le \mathsf d$, we denote
\begin{eqnarray*}\lambda_i(f, \mu)=\int \lambda_i(f,x)\, d\mu(x),\\
\sum_{j=1}^i \lambda^+_j(f, \mu)=\int \sum_{j=1}^i \lambda_j^+(f,x)\, d\mu(x). \end{eqnarray*}
For $\nu\in \mathcal{M}_{erg}(f)$, we have $\lambda_i(f, \nu)= \lambda_i(f,x)$ for all $i$ and for $\nu$-almost every $x$.
By standard arguments the function $\mu\mapsto \sum_{j=1}^i \lambda^+_j(f, \mu)$ defines an affine upper semicontinuous function on $ \mathcal{M}_{inv}(f)$ (see Lemma 3 in \cite{Burguet12}).
For a $\mathcal{C}^{r}$ diffeomorphism with $r>1$ on a compact 3-dimensional Riemannian manifold, we will prove that $E=\frac{\sum_{j=1}^2 \lambda^+_j(f, \cdot)}{r-1}$ satisfies Inequality (\ref{fon}), which together with Theorem \ref{SEX} implies the Main Theorem.
\subsection{Nonuniformly hyperbolic estimates}
Assume now $f$ is a diffeomorphism. In this case, Oseledets theorem provides
for any $\mu\in \mathcal{M}_{inv}(f)$, for $\mu$-a.e., $x\in M$, a
decomposition on the tangent space
$
T_xM=E^{cs}_x\oplus E^{u}_x
$ and $ \rho_{cs}(x)\le 0<\rho_{u}(x)$
satisfying\\
\begin{itemize}
\item
$\underset{|n|\rightarrow
\infty}{\lim}\frac{1}{n}\log\|D_xf^n(v)\|\leq \rho_{cs}(x),\quad \forall\,0\neq v\in E^{cs}_x ;$\\[2mm]
\item
$\underset{|n|\rightarrow
\infty}{\lim}\frac{1}{n}\log\|D_xf^n(w)\|\geq \rho_{u}(x),\quad \forall\,0\neq w\in E^{u}_x ;$
\\[2mm]
\item
$\underset{|n|\rightarrow
\infty}{\lim}\frac{1}{n}\log\sin\angle(E_{f^n(x)}^{cs},E_{f^n(x)}^{u})=0$.
\end{itemize}
For $ 0<\gamma\ll \lambda_{u}$ and $k\in \mathbb{N}$, we consider the sets $\Lambda_k(\lambda_{u}, \gamma)$ consisting of points $x$ in $M$ with the following properties:\\
\begin{itemize} \item
$~ \|Df^n|E_{f^i(x)}^{cs}\|\leq e^{k\gamma
}e^{|i|\gamma}e^{n\gamma
}\,, \quad\forall\, i\in\mathbb{Z}, \,n\geq1;$\\[2mm]
\item
$~
\|Df^{-n}|E_{f^i(x)}^{u}\|\leq e^{k\gamma }e^{|i|\gamma}e^{n(-\lambda_{u}+\gamma)}\,, \quad\forall\, i\in\mathbb{Z}, \,n\geq1;$\\[2mm]
\item
$~
\sin\angle(E_{f^i(x)}^{cs},E_{f^i(x)}^{u})\geq e^{-k\gamma
}e^{-|i|\gamma}\,,\quad \forall\, i\in\mathbb{Z}.$
\end{itemize}
$\,$ \\
From the definition, it holds that \cite{BP, Pollicott} \\
\begin{itemize} \item~ $T_{x}M=E^{cs}_x\oplus E^u_x$ is a continuous splitting on each $\Lambda_k(\lambda_u,\gamma)$; \\[2mm]
\item~
$f^{\pm}(\Lambda_k(\lambda_u,\gamma))\subset \Lambda_{k+1}(\lambda_u,\gamma)$ for any $k\in \mathbb{N}$;\\[2mm]
\item~
$x\in \bigcup_{k\in \mathbb{N}}\Lambda_k( \lambda_{u}, \gamma)$ provided $\lambda_{u}\leq \rho_{u}(x)$;\\[2mm]
\item~ $\lim_{\lambda_u\to 0}\mu\left(\bigcup_{k\in \mathbb{N}} \Lambda_k(\lambda_{u}, \gamma)\right)$ $=1$ for any $\mu\in \mathcal{M}_{inv}(f)$.
\end{itemize}
For the sake of statements, we let $\Lambda_k=\Lambda_k(\lambda_{u}, \gamma)$ for any $k\in \mathbb{N}$ and $\Lambda^*=\bigcup_{k\in \mathbb{N}}\Lambda_k( \lambda_{u}, \gamma)$.
Denote $\lambda'_{u}=\lambda_{u}-2\gamma$. Given $x\in \Lambda^*$, define for all $v=v_{cs}+v_u$ and $w=w_{cs}+w_u$ with $v_{cs},w_{cs}\in E^{cs}_x$ and $v_{u},w_u\in E^{u}_x$,
\begin{eqnarray*}<v_{cs}, w_{cs}>'&=&\sum_{n=0}^{+\infty}e^{-4n\gamma}<D_xf^n(v_{cs}),D_xf^n(w_{cs})>,
\\
<v_{u}, w_{u}>'&=& \sum_{n=0}^{+\infty}e^{2n\lambda'_{u}}<D_xf^{-n}(v_{u}),D_xf^{-n}(w_{u})> ,
\\
<v, w>'&=&<v_{u}, w_{u}>'+<v_{cs}, w_{cs}>'.\end{eqnarray*}
There exists $a_1=a_1(\gamma)>1$ such that
\begin{eqnarray} \label{metric}\|v\|\leq \|v\|'\leq a_1e^{ k\gamma}\|v\|,\quad \forall~v\in T_{\Lambda_k}M.\end{eqnarray}The norm $\|\cdot\|'$ is called a Lyapunov metric, with which $f$ behaves uniformly on $\Lambda^*$:
\begin{eqnarray*}
&&\frac{1}{\|Df^{-1}\|}\|v_{cs}\|'_x \le \|D_xf(v_{cs})\|'_{f(x)}\leq e^{2\gamma}\|v_{cs}\|'_x,\\[2mm]
&& \frac{1}{\|Df\|}\|v_{u}\|'_x \le \|D_xf^{-1}(v_{u})\|'_{f^{-1}(x)}\leq\,e^{-\lambda'_{u}}\|v_{u}\|'_x.
\end{eqnarray*}
In this manner, the splitting $T_{\Lambda^*} M=E^{cs}\oplus E^u$ is dominated with respect to $\|\,\|'$, i.e. $$\frac{\|D_xf(v_{cs})\|'}{\|D_xf(v_u)\|'}\le e^{2\gamma-\lambda_u'}\frac{\|v_{cs}\|'}{\|v_u\|'},\quad \forall\,0\neq v_{cs}\in E^{cs}_x,\,0\neq v_u\in E^u_x,\,\,x\in \Lambda^*,$$
$$\text{with }2\gamma-\lambda_u'<0.$$
We consider a $\mathcal{C}^r$ diffeomorphism $f$ on a $C^r$ smooth Riemanian manifold $(M,\| \cdot \|))$ with $r>1$. Let $\alpha=\min\{r-1,1\}$.
We are going to state that the dominated behavior on each $\Lambda_k$ can be extended to a $e^{-k\mathsf{d}\gamma'}$-neighborhood for $\gamma'=\alpha^{-1}\gamma$. Moreover, for attaining a preassigned local proximity of dominated splitting, we may choose a positive number $b$ independently of $k$ such that this proximity holds in a $be^{-k\mathsf{d}\gamma'}$-neighborhood of $\Lambda_k$.
Let $d$ be the Riemannian distance on $M$ and $\mathsf{r}$ be the radius of injectivity of $(M,\|\cdot\|)$. The ball at $x\in M$ of radius $R\in \mathbb{R}^+$ with respect to $d$ is denoted by $B(x,R)$. Then for $y\in B(x,\mathsf{r})$ we use the identification
\begin{eqnarray*}
T_{B(x,\mathsf{r})}M&\simeq & B(x,\mathsf{r})\times T_xM,\\
(y,v) &\mapsto & \left(y, D_y(\exp_x^{-1})(v)\right)
\end{eqnarray*}
to ``translate" the vector $v\in T_yM$ to the vector $\hat{v}_x:= D_y(\exp_x^{-1})(v)\in
T_xM$.
Recall that the exponential map $(x,v)\mapsto \exp_x(v)$ defines a $C^r$ map (thus $C^{1+\alpha}$) from $TM$ to $M$ with $D_x(\exp_x)=\Id_{T_xM}$. Since the diffeomorphism $f$ is also $C^{1+\alpha}$ on the compact manifold $M$, there
exist $K>1,a_2>0$ such that
\begin{eqnarray*}
\forall x\in M, \ \forall (y,v)\in T_{B(x,a_2)} M, \ & \frac{\|v\|}{2}\leq \|\hat{v}_x\|\leq 2\|v\|\\
& \text{ and } \|D_xf^{\pm}(\hat{v}_x)-\widehat{D_yf^{\pm}v}_{f(x)}\|\leq\,K\|v\|d(x,y)^\alpha.
\end{eqnarray*}
For $x\in \Lambda^*$ and $(y,v)\in T_{B(x,a_2)} M$, we define $\|v\|_x''=\|\hat{v}_x\|'$ and we also let $<,>''_x$ be the associated scalar product on $T_yM$.
It follows then from (\ref{metric}) that
\begin{eqnarray}\label{mettt}\forall x\in \Lambda_k, \ \forall (y,v)\in T_{B(x,a_2)} M, \ \ 2a_1e^{ k\gamma}\|v\|&\geq \|v\|_x''= \|\hat{v}_x\|'&\geq \frac{\|v\|}{2}.\end{eqnarray}
We write $\hat{v}_x$ as $v$
whenever there is no confusion and we also denote by $T_yM=E^{cs}_x\oplus E^{u}_x$ the splitting of $T_yM$ which translates to the splitting $T_xM=E^{cs}_x\oplus E^{u}_x$.
Let $\lambda''_{u}=\lambda'_{u}-\gamma$ and let $a'_2>0$ such that $f^{i}(B(x,a'_2))\subset B(f^{i}x,a_2)$ for all $x\in M$ and $i=0,1,-1$. Then
define
$$\gamma_k=\min\left\{ 1, a'_2, \left(\frac{e^{-\lambda''_u}-e^{-\lambda'_u}}
{4a_1e^{(k+1)\gamma}K}\right)^{\frac{1}{\alpha}}\right\}.$$
Then we have for all $x\in \Lambda_k$, for all $y\in B(x,\gamma_k)$ and for all $v_{cs/u}\in E^{cs/u}_x\subset T_yM$ (see \cite{Pollicott} p.72 for further details) :
\begin{eqnarray}\label{s-con} \Big{(}\frac{1}{\|Df^{-1}\|}-(e^{\gamma}-1)\Big{)} \|v_{cs}\|''_x\le \|D_yf(v_{cs})\|''_{f(x)}\leq e^{3\gamma}\|v_{cs}\|''_x,\\
\label{s-con2} \Big{(} \frac{1}{\|Df\|}-(e^{\gamma}-1)\Big{)} \|v_u\|''_x \le\|D_yf^{-1}(v_u)\|''_{f^{-1}(x)}\leq
e^{-\lambda''_{u}}\|v_u\|''_x.\end{eqnarray}
Define $\kappa(x)=\min\{k\in
\mathbb{N}: x\in \Lambda_k\}$ for $x\in \Lambda^*$. Then the inequalities (\ref{s-con}) and (\ref{s-con2}) hold for any $y\in
B(x,\gamma_{\kappa(x)})$. Such sets $B(x,\gamma_{\kappa(x)})$ are called Lyapunov neighborhoods. Letting $\gamma'=\alpha^{-1}\gamma$, we have $\gamma_k=a_3e^{-k\gamma'}<1$ for $k$ large enough and some constant $a_3$ independent of $k$. We use $d''_x$ to denote the distance induced by $\|\cdot\|_x''$ on $B(x,a_2)$ and $B''_x(y,r)$ to denote the ball centered at $y$ with radius $r$ in $d''_x$.
For the purpose of our use in the computation of tail entropy and local volume growth, we need to estimate the proximity of the dominated splitting in Lyapunov neighborhoods along orbits. For a splitting $F=F_1\oplus F_2$ of an Euclidean space $F$ with norm $\|\,\|$, and $\xi>0$, we denote by $Q_{\|\,\|}(F_1,\xi)$ the cone of width $\xi$ of $F_1$ in $\|\,\|$, i.e. the set $\{v=v_1+v_2\in F:\,v_{1}\in F_1,\,v_2\in F_2,\quad \|v_{2}\|\le \xi \|v_{1}\|\}$. For any vector subspace $G$ of $F$ we let $\iota(G)$ be the Pl\"ucker embedding of $G$ in the projective space $\mathbb{P}\varcurlywedge F$ of the Euclidean power exterior algebra $\varcurlywedge F$. When $A:F\rightarrow F'$ is a linear map between two finite dimensional Euclidean spaces $F$ and $F'$, we let $\varcurlywedge^{l}A$ be the induced map on the $l$-exterior power $\varcurlywedge^l F$ with $l$ less than or equal to the dimension of $F$.
With the above notations the map $x\mapsto\varcurlywedge^l D_x f$ is $\alpha$-H\"older and one may assume its H\"older norm is less than $K$ by taking $K$ larger in advance. Observe that $ \varcurlywedge^l F\ni u \mapsto \|\varcurlywedge^{l}Au \|$ induces a map on $\mathbb{P}\varcurlywedge^l F$ by letting $\|\varcurlywedge^{l}A(\mathbb{P}u) \|=\frac{\|\varcurlywedge^{l}Au \|}{\|u\|}$. Also we let $l_u(f, z)$ be the dimension of $E^u(z)$. When $\mu\in \mathcal M_{erg}(f)$, $l_u(f, z)$ is a constant for $\mu$-a.e. $z$, which we denote by $l_u(f, \mu)$.
\begin{lemma}\label{uniform size}
For any $\xi>0$ small enough there exists $a_\xi>0$ such that for any $x\in \Lambda^*$ and for any $y\in B\left(x,a_\xi\gamma_{\kappa(x)}^{l}\right)$ with $l=l_u(f, x)$ we have :
\begin{itemize} \label{neighborhood}
\item[(i)] $\|D_yf(v)\|''_{f(x)}\geq e^{\lambda''_u-\gamma}\|v\|''_x$ for all $v\in Q_{\|\|_x''}(E^u_x, \xi)$ and $\|D_yf(v)\|''_{f(x)}\leq e^{4\gamma}\|v\|''_x$ for all $v\in Q_{\|\|_x''}(E^{cs}_x, \xi)$, \\[2mm]
\item[(ii)] $D_yf(Q_{\|\|_{x}''}(E^u_x, \xi))\subset Q_{\|\|_{f(x)}''}(E^u_{f(x)},\xi)$ and $D_yf^{-1}(Q_{\|\|_x''}(E^{cs}_x, \xi))\subset Q_{\|\|''_{f^{-1}(x)}}(E^{cs}_{f^{-1}(x)},\xi)$, \\[2mm]
\item[(iii)] $e^{-\gamma }\leq \frac{\|\varcurlywedge^{l}D_yf(\iota(G))\|_{f(x)}''}{\|\varcurlywedge^{l}D_xf(\iota(E^u_{x}))\|_{f(x)}''}\leq e^{\gamma }$ for all $l$-plane $G\subset Q_{\|\|_x''}(E^u_x,\xi)$.
\end{itemize}
\end{lemma}
\begin{proof}Let $\xi>0$ and $x\in \Lambda^*$.\\
\begin{itemize}
\item[(i)] By the domination property $E^{cs}_x\oplus E^u_x $ at $y$ with respect to $\|\cdot \|_x''$ given by the inequalities (\ref{s-con}) and (\ref{s-con2}), the first item holds for small $\xi$ independent of $\kappa(x)$.\\
\item[(ii)] Using the invariance of $E^u$ and the domination property at $x$ there exists $\varsigma\in (0,1)$ independent of $x$ satisfying $D_xf(Q_{\|\|_x''}(E^u_x, \xi))\subset Q_{\|\|_{f(x)}''}(E^u_{f(x)},\varsigma\xi)$. Then for any $y\in B(x,a_\xi\gamma_{\kappa(x)})$, we get by the Inequalities (\ref{mettt})
\begin{eqnarray*}
\|D_xf-D_yf\|_x''&:= &\max_{\|v\|_x''=1}\|D_xf(v)-D_yf(v)\|_{f(x)}''\\ [2mm]
&\leq& 4a_1e^{\kappa\left(f(x)\right)\gamma}\|D_xf-D_yf\| \\ [2mm]
& \leq& 4Ka_1e^{(\kappa(x)+1)\gamma}(a_\xi\gamma_{\kappa(x)})^{\alpha}\\ [2mm]
&\leq & a_\xi^{\alpha}.
\end{eqnarray*}
For small $\gamma$, by (\ref{s-con}) and (\ref{s-con2}) one has also \begin{eqnarray*}
\frac{1}{2\|Df^{-1}\|}\le \min_{\|v\|_x''=1}\|D_yf(v)\|_{f(x)}''\le \max_{\|v\|_x''=1}\|D_yf(v)\|_{f(x)}''
\le 2\|Df\|.
\end{eqnarray*}
It follows that for $\|v\|_x''=1$, the angle $\angle''( D_{y}f(v),D_{x}f(v))$ with respect to $\|\cdot\|_{f(x)}''$ is less than $\arctan(\xi)-\arctan(\varsigma\xi)$ for $a_{\xi}$ small enough. We conclude that $D_yf\left(Q_{\|\|_x''}(E^u_x), \xi\right)\subset Q_{\|\|_{f(x)}''}\left(E^u_{f(x)},\xi\right)$ for any $y\in B(x,a_{\xi}\gamma_k)$. We prove similarly the cone invariance property for the center stable direction. \\
\item[(iii)]To prove the last item observe first that using again the domination property at $x$ we get $\left|\frac{\| \varcurlywedge^{l}D_xf(\iota(G))\|_{f(x)}''}{\|\varcurlywedge^{l}D_xf(\iota(E^u_x))\|_{f(x)}''}-1\right|\leq 1-e^{-\gamma/2}$ for all $l$-planes $G\subset Q_{\|\|_x''}(E^u_x,\xi)$ for $\xi>0$ small enough. As $f$ is $e^{\lambda''_u}$-expanding in the unstable direction with respect to $\|\cdot \|''$ we have $$
\|\varcurlywedge^{l}D_xf(\iota(E^u_x))\|_{f(x)}''\geq e^{l\lambda_u''}.$$
Then arguing as above for $y\in B(x,a_\xi\gamma_{\kappa(x)}^{l})$, we have by Lemma \ref{app} in the Appendix and the Inequalities (\ref{mettt}) :
\begin{align*}\|\varcurlywedge^lD_xf-\varcurlywedge^lD_yf\|_{f(x)}''&\leq (4a_1e^{\kappa\left(f(x)\right)\gamma})^l\|\varcurlywedge^lD_xf-\varcurlywedge^lD_yf\|\leq a_\xi^{\alpha}.\end{align*}
Therefore we get for $a_{\xi}$ small enough :
\begin{align*}\left| \frac{\|\varcurlywedge^{l}D_yf(\iota(G))\|_{f(x)}''}{\|\varcurlywedge^{l}D_xf(\iota(E^u_x))\|_{f(x)}''}-1\right|&\leq \frac{\left|\|\varcurlywedge^{l}D_yf(\iota(G))\|-\|\varcurlywedge^{l}D_xf(\iota(G))\|_{f(x)}''\right|}{\|\varcurlywedge^{l}D_xf(\iota(E^u_x))\|_{f(x)}''}\\[2mm]
&+\left|\frac{\| \varcurlywedge^{l}D_xf(\iota(G))\|_{f(x)}''}{\|\varcurlywedge^{l}D_xf(\iota(E^u_x))\|_{f(x)}''}-1\right|\\
&\leq \frac{a_{\xi}^\alpha}{e^{l\lambda''_u}}+1-e^{-\gamma/2}\\[2mm]
&\leq 1-e^{-\gamma}.
\end{align*}
\end{itemize}
\end{proof}
From the domination structure $E^{cs}\oplus E^{u}$ in the norm $\|\cdot\|''_x$, one may build a family of \textit{fake} center-stable manifolds as follows.
\begin{prop}\label{local manifolds} With the notations of Lemma \ref{neighborhood}, for any $\xi>0$ small enough, there exist $b_{\xi}\in (0,a_\xi)$ and families $\{\mathcal W^{cs}_x:\,x\in \Lambda^*\}$ of $C^1$ manifolds satisfying
\begin{itemize}
\item[(i)] uniform size: for $x\in \Lambda_k$, $k\in \mathbb{N}$,
there is a $C^1$ map $\phi_x:E^{cs}_x\rightarrow E^u_x$ such that $\mathcal W^{cs}_x$ is locally given by the graph $\Gamma\phi_x:=\left\{(z, \phi_x(z)), \ z\in E^{cs}_x\right\}$ of $\phi_x$, i.e. $$\mathcal W^{cs}_x=\exp_x(\Gamma \phi_x)\cap B(x, a_\xi\gamma_k); $$
\item[(ii)] almost tangency: $T_y \mathcal W^{cs}_x$ lies in a cone of width $\xi$ of $E^{cs}_x$ in $\|\cdot\|_x''$ for any $y\in \mathcal W^{cs}_x$;
\\
\item[(iii)] local invariance: $f^{\pm}\mathcal W^{cs}_x(b_\xi\gamma_{\kappa(x)}) \subset \mathcal W^{cs}_{f^{\pm}(x)}$ with $\mathcal W^{cs}_x(\zeta)$ being the ball of radius $\zeta$ centered at $x$ inside $\mathcal W^{cs}_x$ with respect to the distance induced by $\|\cdot\|''_x$ on $\mathcal W^{cs}_x$.
\end{itemize}
\end{prop}
\begin{proof} By taking the exponential map at $x$ we can assume without loss
of generality that we are working in $\mathbb{R}^{\mathsf{d}}$. Let $\xi$ and $a_\xi$ be as in Lemma \ref{neighborhood}. For any $x\in \Lambda^*$, we can extend $f\mid_{B(x,a_{\xi}\gamma_{\kappa(x)})}$ to a diffeomorphism $\tilde{f}_x: \mathbb{R}^{\mathsf d} \to \mathbb{R}^{\mathsf d} $ such that
\begin{itemize}
\item $\tilde{f}_x(y)=f(y)$\quad for $y\in B(x,a_{\xi}\gamma_{\kappa(x)})$;\\
\item $\|D_y\tilde{f}_x-D_xf\|''_x\leq 2 a_{\xi}^{\alpha}$\quad for $y\in \mathbb{R}^{\mathsf d}$.
\end{itemize}
By taking $a_{\xi}$ smaller, the properties of Lemma \ref{neighborhood} hold with respect to $\tilde{f}_x$ for all $y\in \mathbb{R}^{\mathsf d}$. Let $\Xi$ be the disjoint union given by $\Xi=\coprod_{x\in \Lambda^*} \{x\}\times \mathbb{R}^{\mathsf d}$ where $\Lambda^*$ is endowed with the discret topology. Then $\tilde f=(\tilde{f}_x)_{x\in \Lambda^*}$ can be viewed as a map from $\Xi$ to itself by letting $\tilde{f}(x,v)= \left(f(x),\tilde f_x(v)\right)$. Note that the global splitting $\coprod_{x\in \Lambda^*}\{x\}\times \mathbb{R}^{\mathsf d}=\coprod_{x\in \Lambda^*} \{x\}\times (E^{cs}_x\oplus E^u_x)$ is dominated with respect to $\tilde{f}$. By \cite{HPS} $\S5$, we can obtain
a family $ \{ \mathcal{Y}^{cs}_x:\,x\in \Lambda^*\}$ of global $C^1$ submanifolds in $\mathbb{R}^{\mathsf d}$ which are $C^1$ graphs defined on $E^{cs}_x$ such that we have for all $x\in \Lambda^* $ :
\begin{eqnarray*} &&\{x\}\times \mathcal Y^{cs}_x=\bigcap_{n=0}^{+\infty}\tilde{f}^{-n}\left(\{f^n(x)\}\times Q_{\|\|_x''}(E^{cs}_{f^n(x)}, \xi)\right),\\[2mm]
&& \forall y\in \mathbb{R}^{\mathsf d},\ T_y\mathcal Y^{cs}_x \subset Q_{\|\|_x''}(E^{cs}_{x}, \xi).\end{eqnarray*}
In particular we get $\tilde{f}^{\pm}(\{x\}\times \mathcal Y^{cs}_x) \subset\{f^{\pm}(x)\}\times \mathcal Y^{cs}_{f^{\pm}(x)}$. Since we have $\tilde{f}\mid_{\{x\}\times B(x,a_{\xi}\gamma_{\kappa(x)})}=f\mid_{B(x,b_{\xi}\gamma_{\kappa(x)})}$, one concludes the proof by considering $\mathcal W^{cs}_x=\mathcal Y_x\cap B(x, a_{\xi}\gamma_{\kappa(x)})$ and taking much smaller $b_\xi$ than $a_{\xi}$.
\end{proof}
\section{Tail entropy and local volume growth}\label{volume} Let $f:M\rightarrow M$ be a $\mathcal{C}^r$ diffeomorphism with $r>1$ on a compact Riemannian manifold $(M,\|\|)$.
In this section, we relate the Newhouse local entropy of an ergodic measure with the local volume growth of smooth \textit{unstable} discs. We begin with some definitions.
A $\mathcal{C}^{r}$ map $\sigma$, from the unit square $[0,1]^k$ of $\R^k$ to $M$, which is a diffeomorphism onto its image, is called a $\mathcal{C}^r$ $k$-disc. The $\mathcal{C}^r$ size of $\sigma$ is defined as
$$\|\sigma\|_{r}=\sup\{\|D^q\sigma\|:\,\,q\le r,\,\, q\in \mathbb{R}^+\},$$
where $\|D^q\sigma\|$ denotes the $(q-[q])$-H\"older norm of $D^{[q]}\sigma$ for $q\notin \mathbb{N}$ and the usual supremum norm of the derivative $D^q\sigma$ of order $q$ for $q\in \mathbb{N}$.
For any $\mathcal{C}^1$ smooth $k$-disc $\sigma$ and for any $\chi>0$, $1\gg \gamma>0$, $C>1$ and $n\in \mathbb{N}$, we consider the set $\mathcal{H}^n_f(\sigma,\chi,\gamma,C)$ of points of $[0,1]^k$ whose exponential growth of the induced map on the $k$-exterior tangent bundle is almost equal to $\chi$:
$$\mathcal{H}^n_f(\sigma,\chi,\gamma,C):=\left\{t\in [0,1]^k \ : \ \forall 1\leq j\leq n-1, \ C^{-1}e^{(\chi-\gamma)j}\leq \| \varcurlywedge^kD_t\left(f^j\circ\sigma\right)\|\leq Ce^{(\chi+\gamma)j}\right\}.$$
For $\Gamma\subset [0,1]^k$, we also denote by $|\sigma_{|\Gamma}|$ the $k$-volume of $\sigma$ on $\Gamma$, i.e. $|\sigma_{|\Gamma}|=\int_{\Gamma}\|\varcurlywedge^kD_t\sigma\|\, d\lambda(t)$, where $d\lambda$ is the Lebesgue measure on $[0,1]^k$. Then given $\chi>0$, $1\gg\gamma>0$, $C>1$, $x\in M$, $n\in \N$ and $\vep >0$, we define the local volume growth of $\sigma$ at $x$ with respect to these parameters as follows :
$$ V_x^{n,\vep}\left(\sigma \middle| \chi,\gamma,C\right):= \left| (f^{n-1}\circ \sigma)_{|\Delta_n}\right|$$
$$\text{ with } \Delta_n:=\mathcal{H}^n_f(\sigma,\chi,\gamma,C) \cap \sigma^{-1}B_n(x,\vep,f).$$
\begin{prop}\label{ens}
Let $\nu\in \mathcal{M}_{erg}(f)$ with $l=l_u(f, \nu)\geq 1$. Then for any $\varepsilon>0$, $1>\eta>0$ and $\gamma>0$, there exist
a Borel set $F_\eta$ with $\nu(F_\eta)>\eta$ and a constant $C>1$, such that for all $\delta>0$, all $n$ large enough and all $x\in F_{\eta}$ :
\begin{eqnarray*}\label{vvolume}s(n,\delta, B_n(x,\vep,f)\cap F_{\eta})\leq e^{\gamma n}\sup_{\stackrel{\sigma\, l\text{-disk}}{\text{with}\,\|\sigma\|_{r}\le 1}}V_x^{n,2\vep}\left(\sigma \middle| \sum_i\lambda_i^+(\nu),\gamma,C\right).
\end{eqnarray*}
\end{prop}
In fact the $l_{u}$-discs can be chosen to be affine through the exponential map (see the proof of Proposotion \ref{ens} below). Let $v_k^*(f, \vep)$ denote the local volume growth of $k$-disks :
$$v^*_k(f,\vep)=\limsup_n\frac{1}{n}\sup_{x\in M}\sup_{\stackrel{\sigma\, k\text{-disk}}{\text{with}\,\|\sigma\|_{r}\le 1}}\log \left| (f^{n-1}\circ \sigma)_{|\sigma^{-1}B_n(x,\vep,f)}\right|.$$
S. Newhouse \cite{New1, New} proved that the Newhouse local entropy $h^*(f,\nu,\vep)$ of an ergodic measure is less than or equal to the local volume growth of center-unstable dimension. As a direct consequence of Proposition \ref{ens}, we improve this estimate by considering the local volume growth of unstable dimension.
\begin{cor}\label{abo}
With the above notations,
$$\forall \vep>0 \ \forall \nu \in \mathcal{M}_{erg}(f), \ h^*(f,\nu,\vep)\leq v^*_{l_u(f,\nu)}(f,2\vep).$$
\end{cor}
Such an inequality was established by K. Cogswell in \cite{cog} between the Kolmogorov-Sinai entropy and the global volume growth of unstable discs (in particular Cogswell's main result implies Corollary \ref{abo} for $\varepsilon$ larger than the diameter of $M$).
\begin{remark}For any $\nu \in \mathcal{M}_{erg}(f)$, let us denote by $l_{cu}(f,\nu)$ the number of nonnegative Lyapunov exponents of $\nu$. The following estimate is shown in \cite{New} :
\begin{eqnarray*}\hspace{0,8cm} \forall \vep>0 \ \forall\nu \in \mathcal{M}_{erg}(f), \ h^*(f,\nu,\vep)\leq\sup_{\stackrel{\sigma \, l_{cu}(f,\nu)\text{-disk}}{ with \,\|\sigma\|_{r}\le 1}}\limsup_n\frac{1}{n}\sup_{x\in M}\log \left| (f^{n-1}\circ \sigma)_{|\sigma^{-1}B_n(x,2\vep,f)}\right|.\end{eqnarray*}Observe the right-hand
side term differs from the local volume growth $v^*_{l_{cu}(f,\nu)}(f,2\vep)$ as we invert the supremum in $\sigma$ with the limsup in $n$. We do not know if the above inequality still holds true for $l_u$ in place of $l_{cu}$.
\end{remark}
We prove now Proposition \ref{ens} which is the key new tool to prove the existence of symbolic extensions in dimension 3 combining with the approach developed in \cite{Burguet12}.
\begin{proof}[Proof of Proposition \ref{ens}]
Consider $\nu\in \mathcal{M}_{erg}(f)$ with $l=l_u(f,\nu)\geq 1$.
Let $ 0<\gamma\ll \lambda_{u}:=\lambda_{l}(f,\nu)$ in the nonuniformly hyperbolic estimates of Section 2.
Fix $\eta\in (0,1)$ and $k\in \mathbb{N}$ with $\nu(\Lambda_k(\lambda_u,\gamma))>\eta$. There is a subset $F_\eta$ of $\Lambda_k=\Lambda_k(\lambda_u,\gamma)$ with $\nu(F_\eta)>\eta$ such that $\frac{1}{n}\|\varcurlywedge^{l}D_y(f^n|_{E^u_y})\|$ is converging uniformly in $y\in F_\eta$ to $\sum_i\lambda_i^+(f,y)=\sum_i\lambda_i^+(f,\nu)$ when $n$ goes to $+\infty$.
Let $\vep\in (0,1)$ and $\vep_k<\vep$ to be precised.
For any given $\hat x\in F_\eta$, $0<\delta<\vep$, let $E_n$ be a maximal $(n,\delta)$-separated set in $d$ for $f$ in $B_n(\hat x,\vep, f)\cap F_\eta$. There exists $x\in E_n$ such that $E'_n=E_n\cap B(x,\vep_k)$ satisfies $\sharp E'_n \geq A_1\left(\frac{\varepsilon_k}{\varepsilon}\right)^{\mathsf d}\sharp E_n$ for some universal constant $A_1$.
Since we only deal with the local dynamics around the orbit of $x$, we can assume without loss of generality that we are working in $\mathbb{R}^\mathsf{d}$ by taking the exponential map at $x$.
Take $0<\vep_k<(a_1e^{k\gamma})^{-1}$ so small that $B(x, \vep_k)\subset B_x''(x, 2a_1e^{k\gamma}\vep_k) \subset B(x,\vep)$ and consider $$\hat{\mathcal W}_x^{cs}=(x+E^{cs}_x)\cap B''_x(x, 2a_1e^{k\gamma}\vep_k) .$$
For $\theta_{n}=\beta_k e^{-n(4\gamma+l\gamma')}$ with $\beta_k=\beta_k(\delta)$ to be precised we let $\mathcal{A}^{cs}$ be a $\theta_{n}$-net of $\hat{\mathcal W}_x^{cs}$ for $d''_x$ satisfying $\sharp\mathcal{A}^{cs}\le A_2\theta_n^{-\dim E^{cs}}= A_2\theta_n^{-(\mathsf d-l)}$ for some universal constant $A_2$. This means that any point of $\hat{\mathcal W}_x^{cs}$ is within a distance $\theta_n$ of $\mathcal{A}^{cs}$ for $d''_x$. For any $z\in \mathcal{A}^{cs}$, denote $$I_z= \{z+v: \|v\|_x''\le 4a_1e^{k\gamma}\vep_k,\, v\in E_x^u\}.$$
For $y\in B''_x(x,2 a_1e^{k\gamma}\vep_k) $ we let $y=y_{cs}+y_u$ with $y_{cs}\in x+E^{cs}_x$ and $y_u\in E^u_x$. Observe that $E^{cs}_x$ and $E^u_x$ are orthogonal in $<,>_x''$, thus $y_{cs}$ lies in $ \hat{\mathcal W}_x^{cs}$ and there exists $z_y\in \mathcal{A}^{cs}$ with $ \|y_{cs}-z_y\|''_x<\theta_n$. Therefore, when $y$ also lies in $\Lambda_k$ we get :
\begin{align*}
\|y_{cs}-z_y\|''_y&\leq 2a_1e^{k\gamma}\|y_{cs}-z_y\|\\[2mm]
&\leq 4a_1e^{k\gamma}\|y_{cs}-z_y\|''_x\\[2mm]
&\leq 4a_1e^{k\gamma}\theta_n.
\end{align*}
For small $\xi\in(0, \frac{1}{4})$, let $b_{\xi}>0$ be as in Lemma \ref{neighborhood} and Proposition \ref{local manifolds}. Since the distributions $E^{cs}$ and $E^u$ are continuous on $\Lambda_k$, we may choose $\varepsilon_k$ and $\beta_k$ so small that for any $y\in E'_n$:
\begin{itemize}
\item the set $\left([y_{cs},z_y]+E^u_x\right)\cap \mathcal W^{cs}_y$ defines a
graph $\Gamma_{\phi_y}$ of a $C^1$ function $\phi_y: [y_{cs},z_y]\subset E^{cs}_x \rightarrow E^u_x$,
\item $E^{cs/u}_x\subset Q_{\|\|}\left(E^{cs/u}_y,\frac{\xi}{4a_1e^{k\gamma}}\right)\subset Q_{\|\|_y''}\left(E^{cs/u}_y,\xi\right)$, these cones being defined with respect to the splitting $E^{cs}_y\oplus E^u_y$.
\end{itemize}
\begin{figure}[!h]
\begin{tikzpicture}
\draw (0,-1) ellipse (60pt and 30pt);
\filldraw[fill=green!20!white, draw=green!50!black] (-1.5,0.8) .. controls (-2,1.2) and (-1.6,1.3) .. (-1.1,1.5) .. controls (-0.1,1.8) and (0.7,1.7) .. (1.5,1.4)
.. controls(2.3,1.1) and (2,0.9)..(1.4, 0.6)..controls (0.4,0.2) and (-0.7,0.2)..(-1.5, 0.8);
\filldraw [color=blue!100, densely dotted](0.6,0.36) -- (0.6,1.1);
\filldraw [color=blue!100](0.6,-0.7) -- (0.6,0.36);
\filldraw [color=blue!100](0.6,1.1) -- (0.6,2.7);
\filldraw [color=red!100, densely dotted](0.3, 0.31) -- (0.3,1.2);
\filldraw [color=red!100](0.3,-0.6) -- (0.3,0.31);
\draw(0.3,-0.6) -- (0.6,-0.7);
\filldraw [densely dotted](0.3,1.2) -- (0.6,1.1);
\draw (0.3,1.2) ..controls(0.4, 1.17) and (0.55, 1.18).. (0.6,1.2);
\fill (0,-1) circle (1pt);
\node at (-0.2, -1){$x$};
\fill[color=blue] (0.6,-0.7) circle (0.8pt);
\node at (1.2, -0.85){{\small $z_y\in \mathcal{A}^{cs}$}};
\fill [color=red](0.3,-0.6) circle (1pt);
\node at (0, -0.55){{\small $y_{cs}$}};
\fill[color=red] (0.3,1.2) circle (0.8pt);
\node at (0.1, 1.2){{\small $y$}};
\fill[color=blue] (0.6,1.2) circle (0.8pt);
\node at (0.85, 1.3){{\small $t_y$}};
\fill[color=blue] (0.6,1.1) circle (0.8pt);
\node at (2.5, 1.2){{\small $\mathcal W_y^{cs}$}};
\node at (1.7, 2.4){{\small $ I_{z_y}\subset z_y+E^u_x$}};
\node at (2.5, -1){{\small $\hat{\mathcal W}_x^{cs}$}};
\end{tikzpicture}
\caption{The transverse intersection at $t_y
$ of $I_{z_y}$ and $\mathcal W_y^{cs}$ for $y\in E'_n$.}
\end{figure}
Let $\theta_y:[0,1]\rightarrow E^{cs}_x+E^u_x$ be the reparametrization of the graph of $\phi_y$ given by
$$\forall t\in [0,1], \ \theta_y(t)= y_{cs}+t(z_y-y_{cs})+\phi(y_{cs}+t(z_y-y_{cs}).$$
Note that $\theta_y(0)=y$ and $\theta_y(1)$ is the intersection point of $I_{z_y}$ and $\mathcal W_y^{cs}$. To simplify the notations we let $t_y:= \theta_y(1)$. It follows from the almost tangency property of center-stable fake manifolds stated in Proposition \ref{local manifolds} (ii) that
\begin{equation}\label{ein}\theta'(t)\in Q_{\|\|''_y}(E^{cs}_y,\xi).\end{equation}
Moreover we have \begin{eqnarray}
\label{zzwei}z_y-y_{cs}&\in E^{cs}_x \subset Q_{\|\|''_y}(E^{cs}_y,\xi),\\[2mm]
\label{ddrei}D_{y_{cs}+t(z_y-y_{cs})}\phi(z_y-y_{cs})&\in E^u_x\subset Q_{\|\|''_y}(E^{u}_y,\xi).
\end{eqnarray}
From the above properties (\ref{ein}), (\ref{zzwei}), (\ref{ddrei}) and $\xi<\frac{1}{4}$, one deduces after an easy computation that $\|\theta'(t)\|''_y\leq 3 \|z_y-y_{cs}\|''_y$ for all $t\in [0,1]$. For $w\in \Lambda^*$ let $d''_{\mathcal W^{cs}_w}$ be the distance induced respectively by $\|\|''_w$ on $\mathcal W^{cs}_w$. We have
\begin{align*}
d''_{\mathcal W^{cs}_y}(y,t_y)&\leq \int_{[0,1]}\|\theta'(t)\|''_y\, dt,\\[2mm]
&\leq 3\|y_{cs}-z_y\|''_y,\\[2mm]
&\leq 12a_1e^{k\gamma}\theta_n.
\end{align*}
Consequently by the local invariance of center-stable manifolds stated in Proposition \ref{local manifolds} (iii) we get for all $0< j\le n$ :
\begin{eqnarray*}d''_{\mathcal W^{cs}_{f^j(y)}}\left(f^j(y), f^j(t_y)\right)&\le &
\|Df\mid_{T\mathcal W^{cs}_{f^{j-1}(y)}}\|''_{f^{j-1}(y)}
d''_{\mathcal W^{cs}_{f^{j-1}(y)}}\left(f^{j-1}(y),f^{j-1}(t_y)\right),
\end{eqnarray*}
and then by Lemma \ref{neighborhood} (i),
\begin{eqnarray*}
d''_{\mathcal W^{cs}_{f^j(y)}}\left(f^j(y), f^j(t_y)\right)&\le &
e^{4\gamma}
d''_{\mathcal W^{cs}_{f^{j-1}(y)}}\left(f^{j-1}(y),f^{j-1}(t_y)\right).
\end{eqnarray*}
After an immediate induction we obtain for all $0\le j\le n$ :
\begin{eqnarray*}
d''_{\mathcal W^{cs}_{f^j(y)}}\left(f^j(y), f^j(t_y)\right)&\le & e^{4j\gamma}d''_{\mathcal W^{cs}_y}(y,t_y),\end{eqnarray*}
and therefore
\begin{eqnarray*}
d''_{\mathcal W^{cs}_{f^j(y)}}\left(f^j(y), f^j(t_y)\right)
& \le & 12a_1e^{k\gamma} e^{4n\gamma}\theta_n\\[2mm]
&\le & 12a_1e^{k\gamma}\beta_ke^{-nl\gamma'},\\[2mm]
d''_{\mathcal W^{cs}_{f^j(y)}}\left(f^j(y), f^j(t_y)\right)&\le & 12a_1e^{k\gamma}\frac{\beta_k}{\gamma_{\kappa(y)}}\gamma_{\kappa(f^j(y))}.
\end{eqnarray*}
As $y$ belongs to $E'_n\subset \Lambda_k$ we have $\kappa(y)\leq k$. Therefore we get for $\beta_k\leq \frac{b_\xi\gamma_k}{48a_1e^{k\gamma}}$ :
\begin{eqnarray*}
\forall\,0\le j\le n, \ d(f^j(t_y),f^j(y))&\leq& 2 d''_{\mathcal W^{cs}_{f^j(y)}}\left(f^j(y), f^j(t_y)\right)\\[2mm]
&\leq& \frac{b_\xi}{2}\gamma_{\kappa(f^j(y))},
\end{eqnarray*}
and we have similarly for $\beta_k<\frac{\delta}{48a_1e^{k\gamma}} $ :
\begin{align*}
\forall\,0\le j\le n, \ d(f^j(t_y),f^j(y))&\leq \delta/4,\\[2mm]
\text{i.e.} \ t_y \in B_n(y,\delta/4,f)&.
\end{align*}
For $y\in E'_n$ we let now
\begin{eqnarray*} W_n(t_y)&:=&
\bigcap_{j=0}^{n-1}f^{-j}\left(B_{f^j(y)}''(f^{j}(t_y), \frac{\delta}{8} e^{-lj\gamma'})\right)\bigcap I_z,\\[2mm]
&\subset &B_n(t_y,\delta/4,f),\\[2mm]
&\subset & B_n(y,\delta/2,f).
\end{eqnarray*}
As $E'_n$ is $(n,\delta)$-separated, the sets $\left(W_n(t_y)\right)_{y\in E'_n}$ are pairwise disjoint.
For $\delta$ small enough (depending only on $k$), for any $j=0,\cdots, n-1$, the ball $ B_{f^j(y)}''\left(f^{j}(t_y), \frac{\delta}{8} e^{-lj\gamma'}\right)$ is contained in $B\left(f^j(y), b_\xi\gamma_{\kappa(f^j(y))}^{l}\right)$, since $d(f^j(t_y), f^j(y))\le \frac{b_\xi}{2}\gamma_{\kappa(f^j(y))}^{l}$.
Let $(e_x^i)$ be an orthonormal basis of $E^u_x$ with respect to $\|\cdot\|''_x$. We consider the affine reparametrization of $I_z$, $z\in \mathcal{A}^{cs}$, given by $\sigma_z:[0,1]^{l_\nu}\rightarrow M$, $(t_i)_i\mapsto z+\sum_i(t_i-1/2)4a_1e^{k\gamma}\vep_ke_x^i$.
Noting that $E_x^u\in Q_{\|\|_y''}(E^{u}_y, \xi) $, by Lemma \ref{uniform size} (ii), for any $\tau\in \sigma_z^{-1}W_n(t_y)$ and for any $0\leq j\leq n$,
the vector space $D_{\sigma_z(\tau)}f^j(E_x^u)$ lies in $Q_{\|\|''_{f^j(y)}}\left(E^u_{f^j(y)}, \xi\right)$. Then by Lemma \ref{uniform size} (iii)
we get
\begin{eqnarray*}
\limsup_n\frac{1}{n} \log\|\varcurlywedge^{l}D_{\sigma_z(\tau)}f^n|_{E^u_x}\|''_y&=& \limsup_n\frac{1}{n}\sum_{j=0}^{n-1} \log\|\varcurlywedge^{l}D_{f^j\circ\sigma_z(\tau)}f|_{D_{\sigma_z(\tau)}f^j(E_x^u)}\|''_{f^j(y)},\\
&\leq & \limsup_n\frac{1}{n}\sum_{j=0}^{n-1} \log\|\varcurlywedge^{l}D_{f^j(y)}f|_{E^u_{f^j(y)}}\|''_{f^j(y)}+\gamma,\\
&= &\limsup_n\frac{1}{n} \log\|\varcurlywedge^{l}D_{y}f^n|_{E^u_y}\|''_y +\gamma.
\end{eqnarray*}
Noting that $f^n(y)\in \Lambda_{k+n}$, we have by the Inequalities (\ref{mettt})
$$\forall v\in T_{f^n(\sigma_z(\tau))}M, \ \frac{\|v\|}{2}\leq \|v\|''_{f^n(y)}\leq 2a_1e^{(k+n)\gamma}\|v\|.$$
Then it follows from Lemma \ref{app} in the Appendix that
\begin{align*}
\limsup_n\frac{1}{n} \log\|\varcurlywedge^{l}D_{\tau}(f^n\circ \sigma_z)\|&=
\limsup_n\frac{1}{n} \log\|\varcurlywedge^{l}D_{\sigma_z(\tau)}f^n|_{E^u_x}\|,\\[2mm]
&\leq \limsup_n\frac{1}{n} \log\|\varcurlywedge^{l}D_{\sigma_z(\tau)}f^n|_{E^u_x}\|''_y+l\gamma, \\[2mm]
&\leq \limsup_n\frac{1}{n} \log\|\varcurlywedge^{l}(D_{y}f^n|_{E^u_y})\|''_y + (l+1)\gamma,\\[2mm]
&\leq \limsup_n\frac{1}{n}\log \|\varcurlywedge^{l}(D_{y}f^n|_{E^u_y})\| +(2l+1)\gamma,\\[2mm]
&= \sum_i\lambda_i^+(f,\nu)+(2l+1)\gamma.
\end{align*}
Similarly we also get :
$$\liminf_n\frac{1}{n} \log\|\varcurlywedge^{l}D_{\tau}(f^n\circ \sigma_z)\|\geq \sum_i\lambda_i^+(f,\nu)-(2l+1)\gamma.$$
Moreover the above limsup and liminf are uniform in $y\in E'_n$ and $\tau \in \sigma_z^{-1}W_n(t_y)$. Therefore for some $C>1$ we have for $n$ large enough, $$\sigma_z^{-1}W_n(t_y)\subset \mathcal{H}_f^{ n} \left(\sigma_z, \sum_i\lambda_i^+(f,\nu), (2l+2)\gamma, C\right).$$
By using Lemma \ref{uniform size} and classical arguments of graph transform, the set $f^j(W_n(t_y))$ for $0\le j\le n-1 $ defines a graph of a function from $B_{f^j(y)}''\left(f^{j}(t_y),\frac{\delta}{8} e^{-lj\gamma'})\right)\cap E^u_{f^j(y)}$ to $E^{cs}_{f^j(y)}$. Therefore the $l$-volume of $f^{n-1}(W_n(t_y))$ with respect to $\|\cdot \|''_{f^{n-1}(y)}$ satisfies \begin{eqnarray}\label{eins}\left |f^{n-1}(W_n(t_y))\right|''_{f^{n-1}(y)}&\geq & c_{l}\delta^l e^{-{l^2}(n-1)\gamma'},\end{eqnarray}
for some universal constant $c_{l}$. By applying again Lemma \ref{app} in the Appendix we obtain :
\begin{eqnarray}\label{zwei}\left |f^{n-1}(W_n(t_y))\right| &\geq &(4a_1e^{(k+n-1)\gamma})^{-l}\left |f^{n-1}(W_n(t_y))\right|''_{f^{n-1}(y)}
,\end{eqnarray}
where $\left |f^{n-1}(W_n(t_y))\right|$ denotes the $l$-volume of
$f^{n-1}(W_n(t_y))$ with respect to the Riemannian norm $\|\cdot \|$ on $M$. For $z\in \mathcal A^{cs}$ we let $$\Gamma_z:=\{y\in E'_n, \ z_y=z\}$$ and $$\Delta_n^z:= \mathcal{H}_{f}^{n} \left(\sigma_z, \sum_i\lambda_i^+(f,\nu), (2l+2)\gamma, C \right)\cap\sigma_z^{-1} B_n(x, 2\vep, f).$$
As the sets $W_n(t_y)$, $y\in \Gamma_z$, are pairwise disjoint subsets of $\sigma_z(\Delta_n^z)$ we have :
\begin{eqnarray}\label{drei}
\left|(f^{n-1}\circ\sigma_z)_{|\Delta_n^z}\right| &\geq & \sum_{y\in \Gamma_z}\left|f^{n-1}(W_n(t_y))\right|.
\end{eqnarray}
By combining the inequalities (\ref{eins}), (\ref{zwei}), (\ref{drei}) we obtain
\begin{eqnarray*}\left|(f^{n-1}\circ\sigma_z)_{|\Delta_n^z}\right| &\geq & (4a_1e^{(k+n-1)\gamma})^{-l}\sum_{y\in \Gamma_z}\left|f^{n-1}_{|W_n(t_y)}\right|_{f^{n-1}(y)}'',\\
&\geq & c_{l}\delta^l e^{-{l^2}(n-1)\gamma'} (4a_1e^{(k+n-1)\gamma})^{-l}\cdot\sharp\Gamma_z.
\end{eqnarray*} With the notations introduced at the beginning of Section 3, we have therefore for some constant $D$ independent of $n$ and $\hat x\in F_\eta$,
$$\#\Gamma_z \le De^{n(l\gamma+l^2\gamma')} V_x^{n,2\vep}\left(\sigma_z \middle| \sum_i\lambda_i^+(f,\nu), (2l+2)\gamma,C\right). $$
By letting $\mathcal{F}_{n,\delta}:=\{\sigma_z,\ z\in \mathcal{A}^{cs}\}$, we get for all $\hat x\in \Lambda_k$ and some constants, all denoted by $D$ and independent of $n$ and $\hat x\in F_\eta$ :
\begin{eqnarray*} s\left(n,\delta, B_n(\hat x,\vep, f)\right)&=&\sharp E_n,\\[2mm]
&\leq & D\sharp E'_n,\\[2mm]
&\leq & D \sum_{z\in \mathcal{A}^{cs} }\sharp \Gamma_z,\\[2mm]
&\leq &De^{n(l\gamma+l^2\gamma')}\sharp\mathcal{A}^{cs}\sup_{\sigma\in \mathcal{F}_{n,\delta}}V_x^{n,2\vep}\left(\sigma_z \middle| \sum_i\lambda_i^+(f,\nu),(2l+2)\gamma,C\right), \\[2mm]
&\leq & De^{n((\mathsf d-l)(4\gamma+l\gamma')+l\gamma+l^2\gamma')}\sup_{\sigma\in \mathcal{F}_{n,\delta}}V_x^{n,2\vep}\left(\sigma_z \middle| \sum_i\lambda_i^+(f,\nu),(2l+2)\gamma,C\right).
\end{eqnarray*}
This concludes the proof of Proposition \ref{ens} as $\gamma$ and thus $\gamma'=\alpha^{-1}\gamma$ may be chosen arbitrarily small.
\end{proof}
\section{Proof of Main Theorem \ref{maintheorem}}
By Proposition \ref{ens} Newhouse local entropy of an ergodic measure with one positive Lyapunov exponent is bounded from above by the local volume growth of curves. This volume growth may be controled by using the Reparametrization Lemma of \cite{Burguet12}. Following straightforwardly the proof of the Main Proposition in \cite{Burguet12} we get :
\begin{prop}\label{res}Let $f$ be a $\mathcal{C}^r$ diffeomorphism with $r>1$ on a Riemannian manifold $M$ and $\mu\in \mathcal{M}_{inv}(f)$. For all $\gamma>0$, there exist $m_\mu, k_{\mu}\in \mathbb{N}^*$ such that for $\nu\in \mathcal{M}_{erg}(f)$ close enough to $\mu$ with $l_u(f,\nu)=1$, we have
$$h_{m_\mu, k_\mu}^{New}(f, \nu)\leq \frac{\lambda^+_1(f,\mu)-\lambda_1^+(f,\nu)}{r-1}+\gamma.$$
\end{prop}
From the criterion in Theorem \ref{SEX}, for proving the Main Theorem, we need consider all ergodic measures with any possible $l_{u}$.
Actually, the Main Theorem is obtained from the following Proposition by applying Theorem \ref{SEX} with the upper semicontinuous affine function $E:=\frac{1}{r-1}\sum_{i=1,2}\lambda_i^+(f,\cdot)$.
\begin{prop}\label{res2}Let $f$ be a $\mathcal{C}^r$ diffeomorphism with $r>1$ on a 3-dimensional Riemannian manifold $M$ and $\mu\in \mathcal{M}_{inv}(f)$. For all $\gamma>0$, there exist an entropy structure $(h_k)_k$ and $k_\mu\in \mathbb{N}$ such that for $\nu\in \mathcal{M}_{erg}(f)$ close enough to $\mu$, we have
\begin{equation*}\label{12}h_{k_\mu}(f,\nu)\leq \frac{\sum_{i=1,2}\lambda^+_i(f,\mu)-\sum_{i=1,2}\lambda_i^+(f,\nu)}{r-1}+\gamma.\end{equation*}
\end{prop}
In other terms, $E:=\frac{1}{r-1}\sum_{i=1,2}\lambda_i^+(f,\cdot)$ satisfies Inequaliy (\ref{fon}) for a 3-dimensional $\mathcal{C}^r$ diffeomorphism $f$ with $r>1$.
\begin{proof}[Proof of Proposition \ref{res2}]
Fix $\mu\in \mathcal{M}_{inv}(f)$. By the upper semicontinuity of $ \sum_{i=1,2}\lambda^+_i(f,\cdot)$, lower semicontinuity of $\lambda_3(f, \cdot)$ and continuity of the integral of logarithm for Jacobian, when $\nu$ is close enough to $\mu$, one has
\begin{eqnarray}\label{upper} \sum_{i=1,2}\lambda^+_i(f,\mu)- \sum_{i=1,2}\lambda^+_i(f,\nu)&\ge& -\frac{(r-1)\gamma}{2},\\[2mm]
\label{lower} \lambda_3(f,\mu)-\lambda_3(f, \nu)&\le& \frac{\gamma}{2},\\[2mm]
\label{con} \big{|} \int \log \Jac(f)\, d\nu- \int \log \Jac(f)\, d\mu \big{|}&\le& (r-1)\gamma.
\end{eqnarray}
Hence, if $h_{\nu}(f)\le \gamma/2$, from $h_{m, k}^{New}(f,\nu)\le h_{\nu}(f)$ for any $m, k$, by (\ref{upper}) we get
\begin{equation}\label{positive}h_{m, k}^{New}(f,\nu)\leq \frac{\sum_{i=1,2}\lambda_i^+(f,\mu)-\sum_{i=1,2}\lambda_i^+(f,\nu)}{r-1}+\gamma.\end{equation}
Next we assume $h_{\nu}(f)>\gamma/2$.
By Ruelle inequality \cite{Ruelle}, it holds that $\min\left(l_u(f,\nu),l_u(f^{-1}, \nu)\right)=1$. Applying Proposition \ref{res} to $f^{\pm}$, there exist $m_\mu^{\pm}, k_{\mu}^{\pm}\in \mathbb{N}$ such that for any $\nu\in \mathcal{M}_{erg}(f)$ close enough to $\mu$ with $l_u(f^{\pm},\nu)=1$,
$$h_{m_\mu^{\pm}, k_{\mu}^{\pm}}^{New}(f^{\pm},\nu)\leq \frac{\lambda_1^+(f^{\pm},\mu)-\lambda_1^+(f^{\pm},\nu)}{r-1}+\gamma.$$
If $l_u(f,\nu)=1$, then $\sum_{i=1,2}\lambda_i^+(f,\nu)=\lambda_1^+(f,\nu)$, thus by the above inequality, (\ref{positive}) holds with respect to $m_\mu^+, k_{\mu}^+$.
If
$l_u(f^{-1},\nu)=1$, then $\lambda_3(f, \nu)\le -h_{\nu}(f^{-1})=-h_{\nu}(f)<-\gamma/2$, which implies $\lambda_3(f,\mu)<0$ by (\ref{lower}). Thus,
\begin{eqnarray*}
\lambda_1^+(f^{-1},\mu)-\lambda_1^+(f^{-1},\nu)&=& \lambda^-_3(f,\nu)-\lambda^-_3(f,\mu)\\[2mm]
&=&\lambda_3(f,\nu)-\lambda_3(f,\mu).
\end{eqnarray*}
Noting that $ \int \log \Jac(f) \, d\tau =\sum_{i=1,2,3}\lambda_i(f,\tau)$ for any $\tau\in \mathcal{M}_{inv}(f)$, by (\ref{con})
we finally get
\begin{eqnarray*}
\lambda_1^+(f^{-1},\mu)-\lambda_1^+(f^{-1},\nu)&= & \sum_{i=1,2}\lambda_i(f,\mu)-\sum_{i=1,2}\lambda_i(f,\nu)+ \int \log \Jac(f)\, d\nu- \int \log \Jac(f)\, d\mu \\[2mm]
&\le & \sum_{i=1,2}\lambda^+_i(f,\mu)-\sum_{i=1,2}\lambda^+_i(f,\nu)+(r-1)\gamma
\end{eqnarray*}
and therefore
\begin{equation}\label{ddf}
h_{m_\mu^{-}, k_{\mu}^{-}}^{New}(f^{-1},\nu)\leq \frac{\sum_{i=1,2}\lambda^+_i(f,\mu)-\sum_{i=1,2}\lambda^+_i(f,\nu)}{r-1}+\gamma.
\end{equation}
By Lemma 2 in \cite{Burguet12}, the sequence $(\underline{h_k})_k:=(\min(h_{m_\mu^+, k}^{New}(f,\cdot), h_{m_\mu^-,k}^{New}(f^{-1}\cdot)))$ defines an entropy structure. Combining (\ref{positive}) for $l_{u}(f,\nu)= 1$ and (\ref{ddf}) for $l_{u}(f^{-1},\nu)= 1$, we conclude the proof by considering the entropy structure $(\underline{h_k})_k$.
\end{proof}
\begin{remark}
For a local diffeomorphism $f:M\rightarrow M$, the following local Ruelle inequlity holds \cite{BF}\cite{CLY} : there exists a scale $\varepsilon>0$ such that $h^*(f,\mu,\vep)\leq \min\left(\sum_j\lambda^+_j(f,\mu), - \sum_j\lambda^-_j(f,\mu)\right)$ for any $\mu\in \mathcal{M}_{inv}(f)$. In particular in dimension $3$, any invariant measure with positive Newhouse local entropy admits at least one positive and one negative Lyapunov exponent. As the proofs of Main Theorem and Proposition \ref{ens} are just local they apply verbatim in the context of local $3$-dimensional diffeomorphism.
\end{remark}
\appendix
\section{}
Let $E$ and $F$ be two finite dimensional vector spaces of dimension $k$. We endow
$E$ (resp. $F$) with two Euclidean norms $\|\cdot\|_E$ and $\|\cdot\|'_E$ (resp. $\|\cdot\|_F$ and $\|\cdot\|'_F$). We consider the associated Euclidean structures on $\varcurlywedge^k E$ (resp. $\varcurlywedge^k F$). Let $A:E\rightarrow F$ be an invertible linear map and $\varcurlywedge^kA$ the induced map on the $k$-exterior powers. We denote by $\|\varcurlywedge^kA\|$ and $\|\varcurlywedge^kA\|'$ the associated subordinated norms.
\begin{lemma}\label{app}With the above notations. Assume that we have for some constants $C_E,C_F\geq 1$ and $D_E,D_F\leq 1$ :
\begin{eqnarray*}
\forall v\in E, & D_E\|v\|_E \leq \|v\|_E' \leq C_E\|v\|_E,\\
\forall w\in F, & D_F\|w\|_F \leq \|w\|_F' \leq C_F\|w\|_F,
\end{eqnarray*}
then $$(D_F/C_E)^k\|\varcurlywedge^kA\|\leq \|\varcurlywedge^kA\|'\leq (C_F/D_E)^k\|\varcurlywedge^kA\|.$$
\end{lemma}
\begin{proof}
By the singular value decomposition there exists an orthonormal family $(e_i)_{i=1,\cdots, k}$ of $(E, \|\cdot\|_E)$ such that $(Ae_i)_{i}$ is an orthogonal family in $(F,\|\cdot\|_F)$ with $\|\varcurlywedge^kA\|=\|Ae_1 \cdots \wedge Ae_k\|_F=\prod_{i=1}^k\|Ae_i\|_F$.
Similarly we let $(e'_i)_{i=1,\cdots, k}$ be the corresponding orthonormal family for the norms $\|\cdot\|'_E$ and $\|\cdot\|'_F$. Let $P$ be the change of basis matrix from $(e'_i)_i$ to $(e_i)_i$. Then the norms
$\|e_1\wedge\cdots \wedge e_{k}\|_E'$ and $\|e'_1\wedge\cdots \wedge e'_k\|_E$ are just given by the absolute values of the determinants of $P$ and $P^{-1}$ respectively. Therefore we have
\begin{eqnarray*}
|\det(P^{-1})|&\leq &\prod_{i}\|e'_i\|_E\\[2mm]
&\leq& D_{E}^{-k},
\end{eqnarray*}
and
\begin{eqnarray*}
\|e_1\wedge\cdots \wedge e_{k}\|_E'&=&|\det(P)|\\[2mm]
&=&1/|\det(P^{-1})|\\[2mm]
&\geq &D_{E}^{k}.
\end{eqnarray*}
We conclude that
\begin{eqnarray*}
\|\varcurlywedge^kA\|'&\leq &\frac{\|Ae_1\wedge \cdots\wedge Ae_k\|_F'}{\|e_1\wedge \cdots\wedge e_k\|'_E}\\[2mm]
&\leq & D_E^{-k}\prod_i\|Ae_i\|'_F \\[2mm]
&\leq & D_E^{-k}C_F^k\prod_i\|Ae_i\|_F\\[2mm]
&\leq & (C_F/D_E)^k\|\varcurlywedge^kA\|.
\end{eqnarray*}
The other inequality is obtained symmetrically.
\end{proof}
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Career Guidance and Public Policy
Bridging the Gap
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\begin{document}
\author{E. Breuillard and T. Gelander}
\thanks{E.B. acknowledges support from the French CNRS and the IAS Princeton}
\thanks{T.G. was partially supported by NSF grant
DMS-0404557, and by BSF grant 2004010}
\date{\today}
\title{Uniform independence in linear groups}
\begin{abstract}
We show that for any finitely generated group of matrices that is
not virtually solvable, there is an integer $m$ such that, given
an arbitrary finite generating set for the group, one may find two
elements $a$ and $b$ that are both products of at most $m$
generators, such that $a$ and $b$ are free generators of a free
subgroup. This uniformity result improves the original statement
of the Tits alternative.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
The main results of this paper were announced in \cite{note}. We
will say that two elements $x,y$ in a group $\gC$ are {\it
independent} if they satisfy no relation, i.e. if they generate a
non-abelian free subgroup. The classical Tits' alternative
\cite{Tits} says that if $\gC$ is a finitely generated linear
group which is not virtually solvable (i.e. does not contain a
solvable subgroup of finite index), then $\gC$ contains two
independent elements. However, Tits' proof gives no indication of
how deep inside the group one has to look in order to find
independent elements. The main result of this paper is the
following:
\begin{thm}\label{thm1}
Let $\gC$ be a finitely generated non-virtually solvable linear
group. Then there is a constant $m=m(\gC )\in\BN$ such that for
any symmetric generating set $\gS$ ($\gS \ni id$) of $\gC$, there
are two words $W_1,W_2$ of length at most $m$ in the alphabet
$\gS$ for which the corresponding elements in $\gC$ are
independent. In other words, the set $\gS^m$ contains two
independent elements.
\end{thm}
By linear group, we mean any subgroup of $\GL_d(K)$ for some
integer $d\geq 1$ and some field $K$. Let now $\BK$ be a global
field, $\overline{\BK}$ its algebraic closure and $S$ a finite set of places of $\BK$ containing all infinite ones.
We denote by $\OO_\BK(S)$ the ring of $S$--integers.
A subgroup of $GL_d(\overline{\BK})$ will be called
\textit{irreducible} if it does not leave invariant any
non-trivial subspace of $\overline{\BK}^d$ (this is sometimes
called absolutely irreducible). After passing to a suitable
homomorphic image (see Lemma \ref{lem:spec}) of the linear group
under consideration, Theorem \ref{thm1} reduces to the following:
\begin{thm}\label{thm2}
Let $\BK$ be a global field, $S$ a finite set of places of $\BK$
containing all the infinite ones, and $d\geq 2$ an integer. Then
there is a constant $m=m(d,\BK,S)$ with the following property.
Suppose that $\gS\subset\SL_d(\OO_\BK (S))$ is a symmetric subset
containing the identity and generating an irreducible subgroup
whose Zariski closure $\BG$ is semisimple and Zariski connected,
then $\gS^m$ contains two independent elements.
\end{thm}
\begin{rem}
In characteristic zero we can actually find two
independent elements in $\gS^m$ which generate a {\it Zariski
dense} subgroup of $\BG$ (see Theorem \ref{thm:Z-dense} and Remark \ref{rem:Z-dense}).
\end{rem}
As in Tits' original proof we use the classical ping-pong lemma
(Lemma \ref{lem:ping-pong}) for the action of the subgroup
generated by $\gS$ on a projective space over some local field.
Since we are in the arithmetic case, there are only finitely many
candidates for the local field, namely the completions $\BK_v$
with $v \in S$. A substantial part of the proof consists in
finding a ``good" metric on the projective space. If $k$ is a
local field and $H\leq\SL_d(k)$ is a semisimple $k$--subgroup with
corresponding symmetric space (or building) $X$, any point in $X$
determines a metric on $k^d$ hence on the projective space $\BP
(k^d)$. For example, the symmetric space $X=SL_d(\BR)/SO_d(\BR)$
is the space of scalar products on $\BR^d$ with a normalized
volume element. Therefore finding a ``good" metric on $\BP (k^d)$
amounts to finding a ``good" point in $X$.
In Section \ref{sec:displacement}, Lemma \ref{prop:constant-c}, we
establish a useful inequality, a norm-versus-spectrum Comparison
Lemma, that relates the displacement of any finite (and more
generally compact) set $\gS$ of isometries of the symmetric space
(or building) of $\SL_d(k)$ to the displacement of a single
element lying in $\gS^{d^2}$. This Comparison Lemma supplies us
with a good metric on $\BP (k^d)$ and an element in $\gS^{d^2}$
that has a ``large" eigenvalue compared to the Lipschitz constants
(for this good metric) of every generator in $\gS$. With such
information, it is not difficult to produce two proximal elements
with distinct attracting points that will generate a free
semi-group. Hence a consequence of our Comparison Lemma is the
Eskin--Mozes--Oh theorem \cite{EMO} on uniform exponential growth
(for details on this implication and improvements in this
direction, see our subsequent paper \cite{entropy}). However, the
Comparison Lemma alone is not sufficient to prove Theorem
\ref{thm1} and produce the required independent elements. As a
matter of fact, it is usually much harder to generate a free
subgroup than a free semi--group.
In Section \ref{sec:arithmetic-displacement} we prove the
following theorem. Let $\BG$ be a semisimple algebraic
$\BK$--subgroup of $\SL_d$. Let $G=\prod_{v\in S}\BG (\BK_v)$
and $\gC=\BG(\OO_\BK (S))$ be a corresponding $S$--arithmetic
group, which we view as a discrete subgroup of $G$ via the
diagonal embedding. By the Borel Harish-Chandra theorem $\gC$ is a
lattice in $G$, i.e. the quotient space $G/\gC$ carries a finite
$G$--invariant measure. Let $X$ be the product of symmetric spaces
and affine buildings associated to $G$ with a base point $x_0$.
\begin{thm}\label{displacement-growth}
There are positive constants $c_1$ and $c_2$ such that for any
finite subset $\Sigma$ in $\Gamma$ generating a subgroup whose
Zariski closure is connected semisimple and not contained in a
proper parabolic subgroup of $\BG$, we have for all $x \in X$:
\begin{equation*}
d_\gS(x) \geq c_1 d_{X/\gC}(\pi(x),\pi(x_0)) - c_2,
\end{equation*}
where $d_\gS(x)=\max \{ d(g\cdot x,x), g \in \gS \}$, $\pi(x)$ is
the projection of $x$ to the locally symmetric space $X/\gC$ and
$d_{X/\gC}$ is the induced metric on $X/\gC$.
\end{thm}
In other words, the displacement in $X$ of a finite set of lattice
points must grow at a fixed linear rate independently of the
finite set, as one tends to infinity in the locally symmetric
space $X/\gC$, provided that it generates a ``large enough"
subgroup. Note that this theorem is trivial when $\gC$ is uniform.
Moreover, the analogous result holds also for non-arithmetic
lattices (see Remark \ref{rem:n-a-v}) and the constant $c_1$ can
actually be taken to be independent of the choice of the lattice
inside a given group $G$.
At the beginning of the argument proving Theorem
\ref{displacement-growth}, we establish Lemma \ref{k1,l1}, a quantitative version of the Kazhdan--Margulis theorem (namely, if $g\in G$ is "far" from $\gC$ then $g\gC g^{-1}$ contains a non-trivial unipotent ``close" to the identity), which is itself of independent interest.
As a corollary of Theorem \ref{displacement-growth} we obtain
Proposition \ref{prop:gamma}, an arithmetic variant of the
Comparison Lemma. Hence, the outcome of Section
\ref{sec:arithmetic-displacement} is that we can choose the
``good" metric on $\BP (k^d)$ to be arithmetically defined. This
will turn out to be crucial when constructing the ping--pong
players.
Section \ref{sec:ping-pong} is devoted to the construction of the
desired independent elements as ping--pong players on $\BP (k^d)$.
This is done in four steps. First, we construct a proximal
element, second, a very contracting one, third, a very proximal
one, and fourth, a conjugate of the very proximal element which
will form the second ping--pong partner (see Section \ref{prelim}
for this terminology). This construction relies on the study of
the dynamics of projective transformations carried out in
\cite{BG}, and in particular the relation (first used by Tits in
his original proof) between the contraction properties of a
transformation and the Lipschitz constant of its restriction to an
open subset (see Proposition \ref{contracting-properties}). The
arithmetically defined metric that we get from Section
\ref{sec:arithmetic-displacement} supplies us with the two main
ingredients needed to construct the desired ping--pong pair,
namely control on proximality and control on the ability to
separate projective points from projective hyperplanes. The
guiding idea is that the distance between two arithmetically
defined objects is either zero or can be bounded from below by
arithmetic data.
In Section \ref{sec:Z-dense} we restrict to the characteristic $0$ case and show that the bounded independent elements can be chosen to generate a Zariski dense subgroup.
In Section \ref{sec:applications} we describe some consequences of
Theorem \ref{thm1}. One of the main application is that a finitely
generated non--amenable linear group is uniformly non--amenable
and has uniform Cheeger constant, i.e. the family of all Cayley
graphs associated with finite generating sets forms a uniform
family of expanders, see Section \ref{subsec:u-n-a}. One important
consequence is the following:
\begin{thm}\label{thm12345}
Let $\gC$ be a non--virtually solvable linear group. Then there is
a positive constant $\gep$ such that for any finite (not
necessarily symmetric) generating set $\gS$ of $\gC$, and any
finite set $A\subset\gC$, there is some $\gs\in\gS$ for which
$$
\frac{|\gs A\triangle A|}{|A|}>\gep.
$$
\end{thm}
Theorem \ref{thm12345} has several consequences, for instance, for
the growth function of $\gC$ with respect to a varying generating
set. Clearly it implies that $\gC$ has uniform exponential growth,
but in addition it shows that the growth function gets larger when
the generating set get larger. Moreover since in Theorem
\ref{thm12345} we do not assume, in contrast to the situation in
\cite{EMO}, that the generating set $\gS$ is symmetric, we obtain
a uniform exponential growth result for semi--groups rather than
for groups. As another example, note that it implies uniform
exponential growth for spheres rather than for balls. For more
results in this vein see Section \ref{sub-sec:growth}.
We will also show that Theorem \ref{thm1} implies the connected
case of the Topological Tits Alternative from \cite{BG} and
\cite{BG1}. Recall that the connected case of the Topological Tits
Alternative had several interesting consequences such as the
Connes--Sullivan conjecture about amenable actions of subgroups of
real Lie groups, and the Carri\`{e}re conjecture about the
polynomial versus exponential dichotomy for the growth of leaves
in a Riemannian foliation on a compact manifold. In particular,
Theorem \ref{thm1} implies these results as well, see Section
\ref{seb-sec:dense}.
\medskip{Acknowledgements:}
We thank G.A. Margulis for his interest in this work and for many
conversations and discussions, and in particular for suggesting us
Lemma \ref{k1,l1}. We thank A. Salehi--Golsefidy for many
conversations and suggestions which helped to overcome several
difficulties that arose in the positive characteristic case.
\section{Some preliminaries}\label{prelim}
\subsection{Dynamics of projective transformations}
For a more exhaustive and detailed study of the dynamical
properties of projective transformations we refer the reader to
[\cite{BG}, Section 3] and [\cite{BG1}, Section 3].
Let $k$ be a local field and $\left\| \cdot \right\| $ the
standard norm on $ k^{n}$, i.e. the standard Euclidean (resp.
Hermitian) norm when $k$ is ${ \mathbb{R}}$ or ${\mathbb{C}}$ and
$\left\| x\right\| =\max_{1\leq i\leq n}|x_{i}|$ where $x=\sum
x_{i}e_{i}$ when $k$ is non-Archimedean and $ (e_{1},\ldots
,e_{n})$ is the canonical basis of $k^{n}$. This induces an
operator norm on $\text{SL}_{n}(k)$. Consider the standard Cartan
decomposition of $\text{SL}_{n}(k)$,
\begin{equation*}
\text{SL}_{n}(k)=KAK
\end{equation*}
where $K$ is $\text{SO}_{n}(\Bbb{R}),\text{SU}_{n}(\Bbb{C})$ or
$\text{SL}_{n}({\mathcal{O}}_{k})$ according to whether
$k={\mathbb{R}},{\mathbb{C}}$ or is non-Archimedean, and
$A=\{\text{diag}(a_{1},\ldots ,a_{n}):a_{1}\geq \ldots \geq
a_{n}>0,\prod a_{i}=1\}$ if $k$ is Archimedean, and
$A=\{\text{diag} (\pi ^{j_{1}},\ldots ,\pi ^{j_{n}}):j_{i}\in
{\mathbb{Z}},j_{i}\leq j_{i+1},\sum j_{i}=0\}$ if $k$ is
non-Archimedean with uniformizer $\pi $. Any element $g\in
\text{SL}_{n}(k)$ can be decomposed as a product
$g=k_{g}a_{g}k_{g}^{\prime }$, where $k_{g},k_{g}^{\prime }\in K$
and $a_{g}\in A$. The $A$--part $a_{g}$ is uniquely determined by
$g$, but $k_{g},k_{g}^{\prime }$ are not. We will set
\begin{equation*}
a_{g}=\text{diag}(a_{1}(g),\ldots ,a_{n}(g)).
\end{equation*}
Note that $a_{1}(g)=\Vert a(g)\Vert =\Vert g\Vert $. For $g\in
\text{SL} _{n}(k)$ we denote by $[g]$ the corresponding projective
transformation $ [g]\in \text{PSL}_{n}(k)$. Similarly, for $v\in
k^n$ we denote by $[v]$ the corresponding projective point, and
for a linear subspace $H\leq k^n$ we let $[H]$ be the
corresponding projective subspace.
The canonical norm on $k^{n}$ induces the associated canonical
norm on $\bigwedge^{2}k^{n}$. We define the \textit{standard
metric} on $ \Bbb{P}^{n-1}(k)$ by the formula
\begin{equation*}
d\big( [v],[w]\big) =\frac{\left\| v\land w\right\| } {\left\|
v\right\|\cdot\left\| w\right\| }
\end{equation*}
This is well defined and satisfies the following properties:
\textbf{(i)} $d$ is a distance on $\Bbb{P}^{n-1}(k)$ which induces the
canonical topology inherited from the local field $k$.
\textbf{(ii)} $d$ is an ultra--metric distance if $k$ is
non-Archimedean, i.e.
\begin{equation*}
d\big( [v],[w]\big)\leq \max \{d\big( [v],[u]\big) ,d\big( [u],[w]\big)\}
\end{equation*}
for any non-zero vectors $u,v$ and $w$ in $k^{n}$.
\textbf{(iii)} If $f$ is a linear form $k^{n}\to k$, then for any
non-zero vector $v\in k^{n}$,
\begin{equation}
d\big( [v],[\ker f]\big) =\frac{\left| f(v)\right| }{\left\| f\right\| \cdot
\left\| v\right\| } \label{lin}
\end{equation}
\textbf{(iv)} Every projective transformation
$[g]\in\text{PSL}_n(k)$ is bi--Lipschitz on the entire projective
space with Lipschitz constant $|\frac{a_1(g)}{a_n(g)}|^2=\|
g\|^2\cdot\|g^{-1}\|^2$.
\begin{defn}
A projective transformation $[g]\in \text{PGL}_{n}(k)$ is called
\textit{$\epsilon $--contracting}, for some $\epsilon
>0$, if there is a projective hyperplane $[H]$, called a
\textit{repelling hyperplane}, and a projective point $[v]$,
called an \textit{attracting point} such that for all points $[p]\in \Bbb{P}^{n-1}(k)$,
\begin{equation*}
d([p],[H])\geq \epsilon \Rightarrow d([gp],[v])\leq \epsilon .
\end{equation*}
An element $[g]$ is called \textit{$(r,\epsilon )$--proximal}, for
$ r>2\epsilon $, if it is $\epsilon$--contracting with respect to
some $[H],[v] $ with $d([H],[v])\geq r$. An element $[g]$ is
called \textit{$\epsilon$--very contracting} (resp.
\textit{$(r,\epsilon)$--very proximal}) if both $ [g]$ and
$[g^{-1}]$ are $\epsilon$--contracting (resp.
$(r,\epsilon)$--proximal).
\end{defn}
The following proposition summarizes the relations between
contraction, Lipschitz constants and the ratio between the highest
coefficients of $a_g$.
\begin{prop}[See Lemma 3.4 and 3.5 in \cite{BG} and Proposition 3.3 in \cite{BG1}]
\label{contracting-properties} Let $\epsilon \in
(0,\frac{1}{4}],~r\in (0,1]$. Let $g\in \text{SL}_{n}(k)$.
\begin{enumerate}
\item If $|a_{2}(g)/a_{1}(g)|\leq \epsilon $ then $[g]$ is
$\epsilon /r^{2}$--Lipschitz outside the $r$--neighborhood of the
repelling hyperplane $[ \text{span}\{k^{\prime
}{}_{g}^{-1}(e_{i})\}_{i=1}^{n}]$.
\item If the restriction of $[g]$ to some open subset $O\subset {
\mathbb{P}}^{n-1}(k)$ is $\epsilon $--Lipschitz, then $
|a_{2}(g)/a_{1}(g)|\leq \epsilon /2$.
\item If $|a_{2}(g)/a_{1}(g)|\leq \epsilon ^{2}$ then $[g]$ is
$\epsilon $ -contracting, and vice versa, if $[g]$ is $\epsilon
$--contracting, then $ |a_{2}(g)/a_{1}(g)|\leq c\epsilon ^{2}$
where $c$ is some constant depending on $k$.
\end{enumerate}
\end{prop}
Note that the attracting point and repelling hyperplane of a
contracting or proximal element are not uniquely defined. In case
$g$ is semisimple, it is sometimes useful to choose them to be the
span of relevant eigenvectors of $g$, while it is also possible to
define them using the Cartan decomposition like in point $(1)$
above. Very proximal elements are our tool to generate free
subgroups via the following version of the classical ping-pong
lemma:
\begin{lem}[The Ping--Pong Lemma]\label{lem:ping-pong}
Assume that $x$ and $y$ are $(r,\epsilon )$--very proximal
projective transformations of $ \Bbb{P}^{n-1}(k)$ (for some
$r>2\epsilon $), and suppose that the distances between the
attracting points of $x^{\pm 1}$ (resp. of $y^{\pm 1}$) and the
repelling hyperplanes of $y^{\pm 1}$ (resp. of $x^{\pm 1}$) are at
least $r$, then $x$ and $y$ are independent.
\end{lem}
\subsection{How to get out of proper subvarieties in bounded time}
\label{subsection:verities}
\addcontentsline{toc}{subsection}{Bezout's theorem}
Recall the following classical theorem (c.f. \cite{Sch}):
\begin{thm}[Generalized Bezout theorem]\label{thm:Bezout}
Let $\BK$ be a field, and let $X_1,\ldots,X_s$ be pure dimensional
algebraic subvarieties of $\BK^n$. Denote by $Z_1,\ldots,Z_t$ the
irreducible components of $X_1\cap\ldots\cap X_s$. Then
$$
\sum_{i=1}^t \text{deg}(Z_i)\leq \prod_{j=1}^s\text{deg}(X_j).
$$
\end{thm}
For an algebraic variety $X$ we will denote by $\chi (X)$ the sum
of the degrees and dimensions of its irreducible components. The following lemma
is a consequence of Theorem \ref{thm:Bezout} (see Lemma 3.2 in
\cite{EMO} and its proof\footnote{In \cite{EMO} it is assumed that
$\Sigma$ is finite, that the characteristic of the field is $0$,
and that the algebraic group $\BG$ and the variety $X$ are fixed, however the proof in \cite{EMO}
does not depend on these assumptions.}).
\begin{lem}\cite{EMO}\label{Bezout}
Given an integer $\chi$ there is $N=N(\chi)$ such that for any
field $K$, any integer $d\geq 1$, any $K$--algebraic subvariety
$X$ in $GL_d(K)$ with $\chi (X)\leq\chi$ and any subset $\Sigma
\subset {GL_d(K)}$ which contains the identity and generates a
subgroup which is not contained in $X(K)$, we have $\Sigma ^{N}
\nsubseteq X(K)$.
\end{lem}
When $X$ is given, we will sometimes abuse notations and
write $N(X)$ for $N(\chi (X))$.
\section{Reduction to the $S$--arithmetic setting}
Here we reduce Theorem \ref{thm1} to Theorem \ref{thm2}. Given a
global field $\BK$ and a finite set $S$ of places of $\BK$
including all the infinite ones, we denote by $\OO_\BK (S)$ the
ring of $S$--integers in $\BK$. The following lemma is well known:
\begin{lem}\label{lem:spec}
Let $\gC$ be a finitely generated linear group which is not
virtually solvable. Then there is a global field $\BK$, a finite
set of places $S$ of $\BK$ and a representation
$f:\gC'\to\GL_d(\OO_\BK (S))$ of some finite index subgroup
$\gC'\leq\gC$ whose image is Zariski dense in a simple
$\BK$--algebraic group.
\end{lem}
\begin{proof}(Suggested to us by G.A. Margulis)
In the proof of the classical Tits alternative \cite{Tits}, Tits
produces a local field $k$ and a homomorphism $\varphi
:\Gamma\rightarrow\text{GL}_n(k)$ such that $\varphi(\Gamma)$
contains two proximal elements $\varphi
(x),\varphi(y)$ which are ``playing ping--pong" on the projective space ${\mathbb{P}}
^{n-1}(k)$ (i.e. satisfy the hypothesis of Lemma \ref{lem:ping-pong}) and hence generate a free subgroup.
Let $F$ be a global field whose completion is $k$, and let
$\overline{F}$ be its integral closure in $k$, i.e. the field of
all elements in $k$ algebraic over $F$. Let
$X=\text{Hom}(\Gamma,\text{GL}_n(k))$ be the variety of
representations of $\Gamma$ into $\text{GL}_n(k)$, realized as a subset of $
\text{GL}_n(k)^{d(\Gamma )}$ where $d(\gC )$ is the size of some
finite generating set of $\gC$. Then $X$ is an algebraic variety
defined over $F$, and as follows from the implicit function
theorem, the set $X(\overline{F })$ of $\overline{F}$ points is
dense in $X(k)$ in the topology
induced from $\text{GL}_n(k)^{d(\Gamma )}$. Thus we can choose a deformation $
\rho\in X(\overline{F})$ arbitrarily close to $\varphi$. Now if
$\rho$ is sufficiently close to $\varphi$, then $\rho (x)$ and
$\rho (y)$ still play ping--pong on ${\mathbb{P}}^{n-1}(k)$, and
this implies that $\rho (\Gamma )$ is not virtually solvable.
Let ${\mathbb{K}}$ be the field generated by the entries of
$\rho (\Gamma )$. Since $\Gamma$ is finitely generated,
${\mathbb{K}}$ is a global field. Let $\Gamma'$ be a finite
index subgroup of $\gC$ such that $\rho (\Gamma')$ is Zariski connected. We then obtain the representation $
f$ and the group ${\mathbb{G}}$ by dividing by the solvable
radical and projecting to a simple factor of the Zariski closure
of $\rho(\Gamma')$. Note that as $\rho
(\Gamma')\subset\text{GL}_n({\mathbb{K}} )$ its Zariski
closure and solvable radical are defined over ${\mathbb{K}}$. Therefore $
f(\Gamma')\leq{\mathbb{G}} ({\mathbb{K}} )$.
Finally, since $\Gamma$ is finitely generated, there is a finite
set of
places $S$ such that $f(\Gamma )$ lies in the $S$-arithmetic group ${
\mathbb{G}} ({\mathcal{O}}_\BK (S))$.
\end{proof}
It is easy to check that if $n$ is the index of $\gC'$ inside
$\gC$, then for any generating set $\gS\ni 1$ of $\gC$ containing
the identity, $\gS^{2n+1}$ contains a generating set for $\gC'$.
Hence Lemma \ref{lem:spec} implies that Theorem \ref{thm1} is a
consequence of Theorem \ref{thm2}. The main part of this paper is
therefore devoted to the proof of \ref{thm2}.
\section{Minimal norm versus Maximal eigenvalue}
\label{sec:displacement}
In this section, we state and prove the Comparison Lemma, Lemma
\ref{prop:constant-c}. Roughly speaking, this statement says that
the minimal displacement of a compact subset of isometries of a
symmetric space or an affine building is comparable to the minimal
displacement of a single element belonging to some bounded power
of the subset. When we came up with Lemma \ref {prop:constant-c},
we were strongly inspired by Proposition 8.5 in \cite{EMO}.
\subsection{Minimal norm, maximal eigenvalue, and the Comparison Lemma}
Let $k$ be a local field with absolute value $|\cdot |_{k}.$ It
induces the standard norm on $k^{d}$ which in turn gives rise to
an operator norm $\left\| \cdot \right\| $ on $M_{d}(k).$ If $k$
is not Archimedean, let $\mathcal{O}_{k}$ be its ring of integers
and $m_{k}$ the maximal ideal in $\mathcal{O}_{k}.$ We note that
$\left\| a\right\| _{k}\geq 1$ for all $a\in\SL_{d}(k).$ Let
$\Lambda _{k}(a)$ be the maximum absolute value of all eigenvalues
of $a$ (recall that the absolute value has a unique extension to
the algebraic closure of $k$). If $g\in\SL_{d}(k)$ we denote by
$a^{g}$ the conjugate $gag^{-1}$.
For a compact subset $Q\subset M_{d}(k)$ we denote:
\begin{eqnarray*}
\Lambda _{k}(Q) &=&\max \{\Lambda _{k}(a):a\in Q\} \\
\Vert Q\Vert _{k} &=&\max \{\Vert a\Vert _{k}:a\in Q\}~ \\
\gD_{k}(Q) &=&\inf_{g\in \text{SL}_{d}(k)}\Vert gQg^{-1}\Vert
_{k}.
\end{eqnarray*}
\begin{rem}
One can define $\ti \gD_k$ by taking the infimum over
$g\in\PGL_d(k)$. This has some advantages in the non-Archimedean
case, e.g. $\ti\gD_k(Q)=1$ whenever $Q$ lies in a compact group.
Moreover, the ratio between $\ti\gD_k$ and $\gD_k$ is bounded
since $\PSL_d$ has finite index in $\PGL_d$. However, we found it
more convenient for our purposes to use $\gD_k$ as defined above.
\end{rem}
In terms of the action of $\text{SL}_{d}(k)$ on its symmetric
space or affine building, $\log \gD_{k}(Q)$ is, up to a
multiplicative constant, the minimal displacement of $Q$, i.e. the
smallest radius of a $Q$--orbit. When $Q=\{a\}$ is a single
element, diagonalizable over $k$, we have $\gD_{k}(\{a\})=\Lambda
_{k}(a)$. The following gives a similar relation when $Q$ is an
arbitrary compact subset. We denote by $Q^i$ the set of all
products of $i$, not necessarily different, elements of $Q$.
\begin{lem} \label{prop:constant-c} \textbf{(Norm--versus--Spectrum Comparison Lemma)}
There exists a constant $c=c(d,k)>0$ such that for any
compact subset $Q\subset M_{d}(k)$ we have
\begin{equation*}
\gD_{k}(Q)^{i}\geq \Lambda _{k}(Q^{i})\geq c\cdot \gD_{k}(Q)^{i}
\end{equation*}
for some $i\leq d^{2}$.
\end{lem}
\begin{rem}
The proof that we are about to give uses a compactness argument
and hence is not effective. In \cite{entropy} we will give an
effective proof of \ref{prop:constant-c}. This relies on an
effective proof of Wedderburn's theorem on the existence of
idempotents in non-nilpotent subalgebras of matrices.
Additionally, we will show in \cite{entropy} that when $k$ is
non-Archimedean, by taking finite extensions, we can make $c$
arbitrarily close to $1$ (actually $c=(|\pi |_{k})^{2d-1}$), and
derive a strong uniformity result concerning the growth functions
of linear groups.
\end{rem}
We now proceed to the proof of Lemma \ref{prop:constant-c}. We
start with the following classical statement:
\begin{lem}
\label{nilp:span} Let $R$ be a field or a finite ring and let $\mathcal{A}
\leq M_{d}(R)$ be a subring and $R$--submodule. Suppose that
$\mathcal{A}$ is spanned as an $R$--module by nilpotent matrices,
then $\mathcal{A}$ is nilpotent, i.e. $\mathcal{A}^{n}=\{0\}$ for
some $n\geq 1.$
\end{lem}
\begin{proof}
In the $0$ characteristic case, the lemma follows easily from
Engel's theorem using the fact that a matrix is nilpotent iff all
its powers have $0$ trace. The proof we give now works in arbitrary characteristic and
was suggested to us by A. Salehi-Golsefidy.
The ring $\mathcal{A}$ is Artinian and
therefore its Jacobson radical
$J(\mathcal{A})$ is nilpotent. We will prove the lemma by showing
that $\mathcal{A}=J(\mathcal{A})$. Let
$\mathcal{B}=\mathcal{A}/J(\mathcal{A})$ and assume by way of
contradiction that $\mathcal{B}\neq 0$. Now $\mathcal{B}$ is
semisimple, hence by the Artin--Wedderburn theorem,
$\mathcal{B}\cong\bigoplus M_{d_i}(\mathcal{D}_i)$, where the
$\mathcal{D}_i$ are division rings. Since $\mathcal{A}$ is spanned
by nilpotent elements, so is $\mathcal{B}$. This implies that the
trace of any element in $M_{d_i}(\overline{k})$ is $0$, and hence
that $d_i=0$. A contradiction.
\end{proof}
Note that an element $A\in M_{d}(k)$ is nilpotent iff $\Lambda_{k}(A)=0$.
The following generalizes this statement to compact subsets.
\begin{lem}
\label{Lambda&Delta=0} For a compact subset $Q\subset M_{d}(k)$ the
following are equivalent:
$(i)$ $Q$ generates a nilpotent subalgebra.
$(ii)$ $\gD_{k}(Q)=0$.
$(iii)$ $\Lambda _{k}(Q^{i})=0,~\forall i\leq d^{2}$.
\end{lem}
\begin{proof}
Let $\mathcal{A}$ be the algebra generated by $Q$.
\noindent $(i)\Rightarrow (ii)$: By Engel's theorem $\mathcal{A}$
and hence $Q$ can be conjugated by a matrix in SL$_{d}(k)$ into
the algebra of upper triangular matrices with $0$ diagonal.
Conjugating further by some suitable diagonal element in
SL$_{d}(k)$ we can make the norm of $Q$ arbitrarily small.
\noindent $(ii)\Rightarrow (iii)$: For any element $g\in M_d
(k),~\|g\|\geq\Lambda_k(g)$, hence $\gD_k(Q)^i\geq
\gD_k(Q^i)\geq\Lambda_k(Q^i)$.
\noindent $(iii)\Rightarrow (i)$: Take $q\leq d^2$ such that $\dim
(\text{span}\cup_{j=1}^q Q^j)=\dim (\text{span}\cup_{j=1}^{q+1}
Q^{j})$, then $\cup_{j=1}^q Q^j$ spans $\mathcal{A}$. Since
$\Lambda (Q^i)=0$ for $i\leq d^2$, it consists of nilpotent
elements; hence the implication follows from Lemma
\ref{nilp:span}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{prop:constant-c}]
Suppose by contradiction that there is a sequence of compact sets
$Q_1,Q_2,\ldots$ in $M_d(k)$ such that $\Lambda_k(Q_n^i)<
{\gD_k(Q_n^i)}/{n},~\forall i\leq d^2$. By replacing $Q_n$ with a
suitable conjugate of it, we may assume that $\| Q_n\|_k\leq
2\gD_k (Q_n)$, and by normalizing we may assume that $\|
Q_n\|_k=1$. Let $Q$ be a limit of $Q_n$ with respect to the
Hausdorff topology on $M_d(k)$. Then $\| Q\|_k=1$, $\gD_k (Q)\geq
\frac{1}{2}$ since $\gD_k$ is upper semi-continuous, and by
continuity of $\Lambda_k$, $\Lambda_k(Q^i)=0,~\forall i\leq d^2$.
This however contradicts Lemma \ref{Lambda&Delta=0}.
\end{proof}
\subsection{Geometric interpretation of the Comparison Lemma}
For $g\in \text{SL}_{d}(k)$ and $x$ in the associated symmetric
space (resp. affine building) $X$, we denote by $d_{g}(x)=d(g\cdot
x,x)$ the displacement of $g$
at $x$. Similarly, for a compact set $Q\subset \text{SL}_{d}(k)$ we let $
d_{Q}(x)=\max_{g\in Q}d_{g}(x)$. Finally, we consider the minimal
displacement of $g,$ or $Q,$ namely $d_{g}:=\inf_{x\in X}d_{g}(x)$ and $
d_{Q}:=\inf_{x\in X}d_{Q}(x).$
Therefore, Lemma \ref{prop:constant-c} implies the following
geometric statement:
\begin{cor}\label{GeomInt}
There is a universal constant $C=C(d)>0$ such that for any compact set $
Q\subset \text{SL}_{d}(k)$ there exists $g\in \cup _{1\leq i\leq d^{2}}Q^{i}$
such that
\begin{equation*}
\frac{1}{\sqrt{d}}d_{Q}-C\leq d_{g}\leq d^{2}\cdot d_{Q}
\end{equation*}
\end{cor}
\proof Clearly, if $g\in Q^{i}$, $d_{g}\leq d_{Q^{i}}\leq i\cdot
d_{Q}.$ Note that (see Lemma \ref{lem:comp} below) for every $x\in
X,$ and $g\in $SL$_{d}(k),$ we have $\log \left\| g\right\|
_{x}\leq d_{g}(x)\leq \sqrt{d}\log \left\| g\right\| _{x}$ where
$\left\| \cdot \right\| _{x}$ is the norm associated to the
compact stabilizer of $x$ in SL$_{d}(k)$, and the $\log $ is taken
in base $|\pi |_{k}^{-1}$ when $k$ is non-Archimedean. Since the
action of $\SL_d(k)$ on $X$ is transitive in the Archimedean case
and transitive on the cells in the non-Archimedean case, it
follows that $\log \gD_{k}(Q)\geq \frac{1}{\sqrt{d}}(d_{Q}-2)$. On
the other hand, $\log \Lambda _{k}(g)\leq d_{g}$ for all $g\in
$SL$_{d}(k),$ and by Lemma \ref
{prop:constant-c}, there exists an $i\leq d^{2}$ and $g\in Q^{i}$ with $
\Lambda _{k}(g)\geq c\cdot \gD_{k}(Q)^{i}.$ Hence $d_{g}\geq \log
\Lambda _{k}(g)\geq \frac{i}{\sqrt{d}}(d_{Q}-2)+\log c$.
\endproof
\section{Uniform linear growth of displacement functions}\label{sec:arithmetic-displacement}
In this section we prove Theorem \ref{displacement-growth} and
derive an arithmetic analog to Lemma \ref{prop:constant-c} that
will be crucial in the proof of Theorem \ref{thm2}.
Let ${\mathbb{K}}$ be a global field, $S$ a finite set of places
containing all infinite ones and ${\mathcal{O}}_\BK (S)$ the ring of $
S$--integers. For $v\in S$ we let ${\mathbb{K}}_v$ denote the completion of ${
\mathbb{K}}$ with respect to $v$. Since $v$ extends uniquely to
any finite extension of ${\mathbb{K}}_v$ we will, abusing notations, denote by $
|\cdot |_v$ also the absolute value on any such extension. Let
${\mathbb{G}}\leq\SL_d$ be a Zariski connected semisimple
${\mathbb{K}}$--algebraic group.
Let
\begin{equation*}
G=\prod_{v\in S}{\mathbb{G}} ({\mathbb{K}}_v)\leq H=\prod_{v\in S}{\SL_d ({\mathbb{K}}_v)}.
\end{equation*}
The group of $S$--integers ${\mathbb{G}} ({\mathcal{O}}_\BK (S))$
is an $S$--arithmetic group. We will identify it with its diagonal
embedding in $G$. This makes ${\mathbb{G}} ({\mathcal{O}}_\BK
(S))$ a discrete subgroup of $G$. The Borel Harish-Chandra theorem
says that it is a lattice in $G$, i.e. the quotient space
$G/{\mathbb{G}}({\mathcal{O}}_\BK(S))$ carries a finite
$G$--invariant measure, and that if ${\mathbb{G}}$ is
${\mathbb{K}}$--anisotropic then
$G/{\mathbb{G}}({\mathcal{O}}_\BK(S))$ is compact. We will set
$\Gamma ={\mathbb{G}}({\mathcal{O}}_\BK(S))$.
Consider $v\in S$ and set $G_v={\mathbb{G}}
({\mathbb{K}}_v )$ and $H_v=\text{SL}_d({
\mathbb{K}}_v)$. Let $K_v\leq\text{SL}_d({
\mathbb{K}}_v)$ be the maximal compact subgroup corresponding to
the standard norm on ${\mathbb{K}}_v^d$. Recall that for $v$
Archimedean any maximal compact is conjugate to $K_v$ in
$\text{SL}_d({\mathbb{K}}_v)$, and for $v$ non-Archimedean there
are $d+1$ conjugacy classes. Let
$X_v=\text{SL}_d({\mathbb{K}}_v)/K_v$ be the
associated symmetric space or affine building, let $A_v$ be a Cartan semigroup of $\text{SL}_d({
\mathbb{K}}_v)$ corresponding to $K_v$ with respect to a Cartan
decomposition of $\text{SL}_d({\mathbb{K}}_v)$, and let $x_0\in
X_v$ be the point corresponding to $K_v$.
We also set the following notations. For $a\in \SL_d({\mathbb{K}})$ let
\begin{equation*}
\Lambda (a)=\max \{|\lambda |_{v}:\lambda \text{~is an eigenvalue of~}
a,~v\in S\}=\max \{\Lambda _{v}(a):v\in S\}.
\end{equation*}
For $v\in S$ let $\Vert \cdot \Vert _{v}$ be the standard operator norm on $\text{SL}_{d}(
{\mathbb{K}}_{v})$, and for $g=(g_{v})_{v\in S}\in H$ let
\begin{equation*}
\Vert g\Vert =\max \{\Vert g_{v}\Vert _{v}:v\in S\}.
\end{equation*}
For a compact subset $Q\subset H$ we let
\begin{equation*}
\Vert Q\Vert =\max_{a\in Q}\Vert a\Vert ,~~\gD(Q)=\inf_{h\in
H}\Vert Q^{h}\Vert=\max_{v\in S}\Delta_{\BK_{v}}(Q)
,~~\text{and}~~\Lambda (Q)=\max_{a\in Q}\Lambda (a)=\max_{v\in
S}\Lambda_{\BK_{v}}(Q).
\end{equation*}
\subsection{Restating Theorem \ref{displacement-growth}}
\begin{defn}
We will say that a subgroup of $\BG$ is irreducible in $\BG$ if
it is not contained in a proper parabolic subgroup of $\BG$.
\end{defn}
Recall the following result of Mostow in the Archimedean case
(c.f. \cite {Pla-Rap} Theorem 3.7) and Landvogt in the
non-Archimedean (c.f. \cite{Landvogt}):
\begin{thm}
\label{thm:ML} There exists a point $x_1\in X_v$ when $v$ is
Archimedean (resp. a cell $\gs_1\subset X_v$ when $v$ is
non-Archimedean) such that the orbit $G_v\cdot x_0$ (resp.
$\cup\{g\cdot\gs:{g\in G_v}\}$) is convex and isometric to the
symmetric space (resp. affine building) associated to $G_v$.
\end{thm}
Recall that the norm of a matrix in $\text{SL}_d$ is comparable to
the exponent of its displacement. More precisely:
\begin{lem}
\label{lem:comp}For any $h\in \text{SL}_{d}({\mathbb{K}}_{v})$ we
have:
\begin{itemize}
\item $\Vert h\Vert \leq e^{d(h\cdot x_{0},x_{0})}\leq \Vert h\Vert ^{\sqrt{
d}}$.
\item If $x=g^{-1}\cdot x_{0}$ then $\Vert h^{g}\Vert \leq
e^{d(h\cdot x,x)}\leq \Vert h^{g}\Vert ^{\sqrt{d}}$.
\end{itemize}
\end{lem}
\begin{proof}
If $h =k_h a_h k_h'$ is a $KAK$ expression for $h$ then $\| h\|
=\| a_h\|$ and $d(h\cdot x_0,x_0)=d(a_h\cdot x_0,x_0)$ hence its
enough to prove the first inequality for elements in $A$, and for
such elements it follows by a direct computation.
The second inequality is a direct consequence of the first one.
\end{proof}
Assume that $\BG$ is isotropic over $\BK$, i.e that $\Gamma$ is a
non-uniform lattice in $G$. Let $\pi :G\to G/\Gamma$ be the
canonical projection, and for $g\in G$ denote
\begin{equation*}
\|\pi (g)\|=\min_{\gamma\in\Gamma}\| g\gamma\|.
\end{equation*}
Note that the convex orbit of $G$ from Theorem \ref{thm:ML} may
not pass through the origin $x_0$, however, since any two orbits
of $G$ are equidistant, in view of Lemma \ref{lem:comp}, Theorem
\ref{displacement-growth} can be restated as follows:
\begin{thm}\label{thm:DG}
There are positive constants $C_1,C_2$ such that for any finite subset $\Sigma$
in $\Gamma$ generating a subgroup whose Zariski closure is
semisimple and irreducible in $\BG$, we have $\forall g\in G$
\begin{equation}
\|\gS^g\|\geq C_2\|\pi(g)\|^{C_1}.
\end{equation}
\end{thm}
\begin{rem}\label{rem:n-a-v}
The statement of Theorem \ref{thm:DG}, as well as of Lemma
\ref{k1,l1} below, remains true without the assumption that the
non-uniform lattice $\gC$ is arithmetic.
To see this one carries the
same argument as below, using a variant of Corollary 8.16 from
\cite{raghunathan} instead of Lemma \ref{epsilon}. Moreover, the constant $C_1$ can be taken to depend only on $G$
and not on the choice of the lattice $\gC$.
\end{rem}
\subsection{A quantitative Kazhdan--Margulis Theorem }
Let $\BK,S,\BG,G,\gC$ be as in the previous paragraph, in
particular we assume that $G/\gC$ is non-compact.
According to the Kazhdan--Margulis Theorem (see
\cite{raghunathan}), if $\|\pi (g)\|$ is large enough, then
$\Gamma ^{g}$ contains a non-trivial unipotent close to the
identity. The following quantitative version of this theorem was
suggested to us by G.A. Margulis.
\begin{lem}
\label{k1,l1} There are positive constants $k_{\gC},l_{\Gamma }$
such for any $g\in
G$ the lattice $\Gamma ^{g}=g\gC g^{-1}$ contains a non-trivial
unipotent $u\in \Gamma ^{g}$ with
\begin{equation*}
\Vert u-1\Vert \leq l_{\Gamma }\Vert \pi (g)\Vert ^{-k_{\gC}}.
\end{equation*}
\end{lem}
Lemma \ref{k1,l1} is proved along the same lines as the original
Kazhdan-Margulis Theorem.
\begin{lem}
\label{epsilon} There is a positive constant $\epsilon _{G}$ such that if $u_{1},\ldots ,u_{t}$ are elements
belonging to a non-uniform $S$--arithmetic subgroup of $G$ and
$\Vert u_{i}-1\Vert \leq \epsilon _{G},~\forall i\leq t$, then the
group $\langle u_{1},\ldots ,u_{t}\rangle $ is unipotent.
\end{lem}
\begin{proof}[Proof of Lemma \ref{epsilon}]
If $\gep_G$ is sufficiently small then by the Zassenhaus Lemma
(c.f. \cite{raghunathan} 8.8. and 8.17.) the $u_i$'s generate a
nilpotent group, and by \cite{margulis} 4.21(A) the $u_i$ are
unipotent. The result follows since any nilpotent group which is
generated by unipotent elements is unipotent.
\end{proof}
\begin{proof}[Proof of Lemma \ref{k1,l1}]
We will first assume that $\text{char}({\BK})=0$, and later
indicate the changes to be made in the positive characteristic
case.
For any Zariski connected unipotent group $U$ there is an element
$g_U\in G$ such that the restriction of $\text{Ad}(g_U)$ to the
Lie algebra of $U$ expands the norm of any element by at least a
factor $4$. Since the Grassmann manifolds are compact, it follows
that there are finitely many elements $g_1,\ldots
,g_k,~g_i=g_{U_i}$ such that for any Lie subalgebra
$\mathfrak{u}$, corresponding to some unipotent subgroup, there is
$i\leq k$ such that the restriction of $\text{Ad}(g_i)$ to
$\mathfrak{u}$ expands the norm of any element by at least a
factor $3$. Now since the exponential map $\exp :\text{Lie}(G)\to
G$ is a diffeomorphism near the origin $0$ of Lie$(G)$ with
differential $1$ at $0$ it follows that for some $\gep_1>0$,
smaller than $\gep_G$, we have:
\begin{equation}
\|g_iug_i^{-1}-1\|\geq 2\| u-1\|,~~\forall~u\in\exp (\mathfrak{u})
~\text{with}~\|u-1\|\leq\gep_1.
\label{star}
\end{equation}
Fix
$$
a=\max_{i\leq k}\| g_i^{\pm 1}\|.
$$
and let
$$
k_{\gC}=\log_a 2.
$$
Fix $\gep_2>0$ smaller than $\gep_1$, such that
$$
B_{\gep_2}(1_G)\subset\cap_{i\leq k}(B_{\gep_1}(1_G))^{g_i^{\pm
1}}.
$$
Since $M=G/\gC$ has finite volume the ``$\gep_2$--thick part"
$$
M_{\geq\gep_2}:=\{\pi (g):g\in G,~\text{and}~\gC^g\cap
B_{\gep_2}(1_G)=\{ 1\}\}
$$
is compact. Let
$$
l_\gC=\sup_{\{ g: \pi (g)\in M_{\geq\gep_2}\}}\big(\min_{\text{all
unipotents}~u\in\gC\setminus\{ 1\}}\| u^g-1\|\big)\cdot \sup_{\{ g: \pi (g)\in
M_{\geq\gep_2}\}}(\|\pi (g)\|)^{k_{\gC}},
$$
then the lemma holds for any $g$ with $\pi (g)\in M_{\geq\gep_2}$.
Now suppose $\pi (g)\notin M_{\geq\gep_2}$. Then $\gC^g$ has a
non-trivial unipotent in $B_{\gep_2}(1_G)$. Let
$$
b=\min \{\| u^g-1\| :u\in\gC\setminus\{ 1\}~\text{unipotent}\}.
$$
By Lemma \ref{epsilon} $\gC^g\cap B_{\gep_1}(1)$ is contained in
Zariski connected unipotent group, and hence by $(\ref{star})$
there is some $g_{i_1}$ such that the conjugation by it increases
the distance of any non-trivial element of this intersection by at
least a factor of $2$. After this conjugation, there might be some
new unipotent elements in the $\gep_1$--ball around $1_G$,
however, by the choice of $\gep_2$ there are no new unipotents in
the $\gep_2$--ball. Therefore we can iterate this argument
$\lceil\log_2\frac{\gep_2}{b}\rceil$ times, and get a sequence
$g_{i_1},\ldots,g_{i_t},~t={\lceil\log_2\frac{\gep_2}{b}\rceil}$,
such that $\gC^{g_{i_{t}}\cdot\ldots\cdot g_{i_1}g}$ intersect
$B_{\gep_2}(1_G)$ trivially. It follows that $\pi
(g_{i_{t}}\cdot\ldots\cdot g_{i_1}g)\in M_{\geq\gep_2}$, and
hence, if $u\in\gC^{g_{i_{t}}\cdot\ldots\cdot g_{i_1}g}$ is a
non-trivial unipotent closest to $1_G$
$$
\|\pi (g_{i_{t}}\cdot\ldots\cdot g_{i_1}g)\|^{k_{\gC}}\cdot\|
u-1\|\leq l_\gC.
$$
Since $\| u-1\|\geq 2^tb$, and since all the $g_i$'s have norm at
most $a$, the result follows.
\medskip
Let us now explain the required modifications in the proof for the
positive characteristic case. For the positive characteristic
version of the Kazhdan--Margulis theorem see \cite{raghunathan1}.
In the positive characteristic case, the unipotent group provided
by Lemma \ref{epsilon} is not Zariski connected, in fact it is
finite. However, it was shown by Borel and Tits \cite{borel-tits}
that for any unipotent group $U$ there is a canonical parabolic
group $P(U)$ which contains the normalizer of $U$ and contains $U$
in its unipotent radical. The unipotent radical of a parabolic
subgroup is Zariski connected, and pro-$p$. Using the $KP$
decomposition where $P$ is a minimal parabolic and $K$ is a
maximal compact, it is easy to show that there is some compact set
$C$ such that for any parabolic subgroup there is $g\in C$ such
that conjugation by $g$ expends the norm of each element in the
unipotent radical of the parabolic by at least $4$, and one can
carry out the same argument as above.
\end{proof}
\subsection{Proof of Theorem \ref{thm:DG}}
Let $\gC,\BG$, $G$, $k_{\gC}$ and $l_{\gC}$ be as in the previous paragraph. Clearly, the
following claim implies Theorem \ref{thm:DG}:
\begin{clm}\label{lem:eq3}
There is a constant $N$, depending only on $\BG$, such that
$\forall g\in G$
\begin{equation}
l_\gC\|\pi(g)\|^{-k_{\gC}}\|\Sigma^g\|^{2N}\geq\epsilon_G.
\label{estimate pi(g)}
\end{equation}
\end{clm}
\begin{proof}
Assume first that char$({\mathbb{K}} )=0$ and let $N=d^2$. Suppose by
way of contradiction that the lemma is false, and let
$u\in\Gamma^g\setminus\{ 1\}$ be a unipotent element as in Lemma
\ref{k1,l1} with
\begin{equation*}\label{eq18}
\|u-1\|\leq l_\gC\|\pi (g)\|^{-k_{\gC}}.
\end{equation*}
Then it follows that for any word $W$ in the elements of
$\Sigma^g$ of length at most $d^2$ we have $\|
u^W-1\|\le\epsilon_G$. Let $\mathcal{U}_i$ be the Zariski closure
of the group generated by $\{ u^W:W$ is a word in the
elements of $\Sigma^g$ of length $\leq i\}$. Then by Lemma \ref{epsilon}, $
\mathcal{U}_i$ is a unipotent group, hence is Zariski connected.
Therefore,
for some $i_0<\dim ({\mathbb{G}} )\leq d^2$ we have $\mathcal{U}_{i_0}=
\mathcal{U}_{i_0+1}$, and hence $\mathcal{U}_{i_0}$ is normalized by $
\Sigma^g$. But this implies that $\Sigma^g$ is contained in some
proper parabolic subgroup, a contradiction to the assumption that
$\Sigma$ generates an irreducible subgroup. Hence the claim is
proved.
We now give an alternative proof which holds in arbitrary
characteristic. Let $U$ be a maximal unipotent subgroup of $\BG$.
For any $u\in U\setminus\{1\}$ let $Y_u=\{ h\in\BG:u^h\in U\}$.
Then $Y_u$ is a proper algebraic subset of $\BG$, and one easily
sees that $\chi (Y_u)$ is bounded independently of $u$, by some $\chi$ say. Now if $\BE$
is an irreducible subgroup of $\BG$, i.e. not contained in a proper
parabolic subgroup, then $\cap_{h\in \BE}U^h$ is trivial. It follows
that $\BE\varsubsetneq Y_u$ for any $u\in U\setminus\{1\}$.
Thus Lemma \ref{Bezout} yields a constant
$N=N(\chi)$ such that
some word $W$ of length at
most $N$ in $\Sigma^g$ satisfies $\| u^W-1\|>\epsilon_G$, where $u$ is the element in $(\ref{eq18})$. For
otherwise, $\langle u^W:W\text{~is a word of length~}\leq
N\text{~in~}\Sigma^g\rangle$ would be a unipotent group and hence
some conjugate of it would lie in $U$, and since the corresponding
conjugate of $\Sigma^g$ generates a group whose Zariski closure
$\BE$ is irreducible in $\BG$, this contradicts the property of
$N(\chi)$. It follows that equation (\ref{estimate pi(g)}) holds
with $N=N(\chi)$.
\end{proof}
\subsection{An $S$--arithmetic version of the Comparison Lemma}\label{subMini}
Let $\BK,S,\BG,G,\gC,d$ be as in the beginning of this Section (we
do not assume that $\BG$ is isotropic over $\BK$). The goal of the
remaining part of this section is to prove the following
arithmetic version of Lemma \ref{prop:constant-c}.
\begin{prop}
\label{prop:gamma}\label{thm:gamma} For some constant $r$,
depending only on $\BG$, $\BK$, and $S$,
we
have that for any finite subset $\Sigma \subset \Gamma={\mathbb{G}}({\mathcal{O}}_\BK(S)) $ ($\Sigma
\ni id$) generating a subgroup whose Zariski closure $\BF$ is
irreducible in $\BG$, there is an element $\gamma \in
\Gamma $ such that $\Vert \Sigma ^{\gamma }\Vert \leq \Lambda
(\Sigma ^{r})$.
\end{prop}
\begin{rem}
In the proof of Theorem \ref{thm2} in the next section we will
apply Proposition \ref{thm:gamma} only in the case where
$\BG=\SL_d$ and $\gC=\SL_d(\OO_\BK(S))$.
\end{rem}
Note that if $\alpha\in{\mathbb{G}}({\mathcal{O}}_\BK (S))$,
$\Lambda (\alpha )=1$ if and only if all the eigenvalues of
$\alpha$ are roots of unity, i.e. if and only if $\alpha$ has
finite order. Moreover, there is a positive constant $\tau>1$ such
that if $\alpha\in{\mathbb{G}}({\mathcal{O}}_\BK (S))$ has
$\Lambda (\alpha )>1$
then $\Lambda (\alpha )\geq\tau$. This follows from the fact that ${\mathcal{O
}}_\BK (S)$ embeds discretely in $\prod_{v\in S}{\mathbb{K}}_v$.
Moreover, the Zariski closure $Y$ of the set of torsion elements
in $\Gamma$ is a proper algebraic subvariety of $\BG$ (there is an
upper bound of the order of torsion elements in $\Gamma$, see
Proposition 2.5. \cite{Tits}).
Hence Lemma \ref{Bezout} implies that $\Sigma^{\overline{r}}$
contains a non torsion element, where $\overline{r}$ is some
integer independent of $\gS$.
This shows that $\Lambda(\Sigma^{n})\geq \tau$ for all
$n\geq\overline{r}$. We can therefore reformulate the Comparison
Lemma (\ref{prop:constant-c}) as follows, omitting the
multiplicative constant.
\begin{lem}
\label{r1} For some constant $r'$, depending only on $\BG$, $\BK$, and $S$,
we
have that for any finite subset $\Sigma \subset \Gamma={\mathbb{G}}({\mathcal{O}}_\BK(S)) $ ($\Sigma
\ni id$) generating a subgroup whose Zariski closure $\BF$ is
irreducible in $\BG$, there is an element $h\in H$ such that
$\Vert \Sigma ^{h}\Vert \leq \Lambda (\Sigma ^{r^{\prime }})$.
\end{lem}
In order to derive Proposition \ref{prop:gamma} from Lemma
\ref{r1} we will first replace the conjugating element $h\in H$ by
an element $g\in G$ (of course this step is unnecessary when $G=H$
which is the situation in the proof of Theorem \ref{thm2}). The
second part of the proof which consists in replacing $g$ by some
$\gamma\in\Gamma$ relies on Theorem \ref{thm:DG}.
\medskip
\subsubsection{Step 1: Projection to a homogeneous subspace}
By Theorem \ref{thm:ML} we may identify the symmetric space (resp. affine building) of
$G_v$ with a convex subset $C$ of $X_v$ of the form $G_v\cdot x_1$
(resp. $G_v\cdot\gs_1$) for some point $x_1$ (resp. some cell $\gs_1 \ni x_1$) in $X_v$.
Since $X_v$ is a CAT(0) space, the projection to the nearest point
$P_C:X_v\to C$ is $1$--Lipschitz. Let
$h\in\text{SL}_d({\mathbb{K}}_v)$ be the element from Lemma
\ref{r1}, let $x=P_C(h\cdot x_0)$ and let $g_v\in G_v$ be an
element such that $g_v\cdot x=x_1$ (resp. $g_v\cdot x\in \gs_1$). In
any case, we have $d(x_1,g_v\cdot x)\leq 1$. Since $\gS\subset
G_v$, it preserves $C$ and since $P_C$ is $1$--Lipschitz we have
$d_\gS(x)\leq d_\gS(h\cdot x_0)$, where
$d_\gS(x)=\max_{\gc\in\gS}d(x,\gc\cdot x)$. We get $d_\gS (g_v^{-1}\cdot x_1)\leq
d_\gS(h\cdot x_0)+2$, and finally we obtain:
$$
d_\gS (g_v^{-1}\cdot x_0)\leq
d_\gS(h\cdot x_0)+2+2d(x_0,x_1).
$$
With Lemma \ref{lem:comp} we can translate this to:
$\|\gS^{g_v}\|\leq\|\gS^h\|^b$ for some constant $b>0$. Repeating
this argument for every $v \in S$, we get from Lemma \ref{r1}:
\begin{cor}
\label{r2} For some constant $r^{\prime \prime }$ (independent of $\gS$) we have
\begin{equation*}
\Vert \Sigma ^{g}\Vert \leq \Lambda (\Sigma ^{r^{\prime \prime
}}).
\end{equation*}
\end{cor}
\subsubsection{Step 2: Finding a relatively close point in a given
$\gC$--orbit} We will now explain how to replace $g=(g_v)\in G$
by some $\gc\in\gC$ and obtain the proof of Proposition
\ref{prop:gamma}.
Assume first that ${\mathbb{G}}$ is ${\mathbb{K}}$--anisotropic, i.e. that $
G/\Gamma$ is compact. Let $\Omega$ be a fixed bounded fundamental
domain for
$\Gamma$ in $G$ and let $\gamma\in \Gamma$ be the unique element such that $
g\in \Omega\gc$. Write
\begin{equation*}
c=\max \{\| f\| :f\in \Omega\cup\Omega^{-1}\},
\end{equation*}
then Theorem \ref{prop:gamma} holds with $r=r''(1+2\log_{\tau}c)$.
Next assume that $\BG$ is $\BK$--isotropic. By equation
(\ref{estimate pi(g)})
\begin{equation*}
\|\pi(g)\|\leq \big( l_\gC\frac{\|\Sigma^g\|^{2N}}{\epsilon_G}\big)
^{1/k_{\gC}}\leq \big( l_\gC\frac{\Lambda (\Sigma^{r''})^{2N}}{\epsilon_G}\big)
^{1/k_{\gC}}.
\end{equation*}
Which means that for some $\gamma\in\Gamma$
\begin{equation*}
\|g\gamma^{-1}\|\leq \big( l_\gC\frac{\Lambda (\Sigma^{r''})^{2N}}{\epsilon_G}
\big)^{1/k_{\gC}} \leq \Lambda (\Sigma^{r''} )^{\frac{2N}{k_{\gC}}+\frac{1}{k_{\gC}}\log_\gt\frac{
l_\gC}{\epsilon_G}}.
\end{equation*}
and therefore
\begin{equation*}
\|\Sigma^\gc\|=\|\Sigma^{\gamma
g^{-1}g}\|\leq\|\Sigma^g\|\|g\gamma^{-1}\|\|\gamma g^{-1}\|\leq \Lambda
(\Sigma^r)
\end{equation*}
for some computable constant $r$. \qed
\section{Construction of the ping-pong players}\label{sec:ping-pong}
In this section we will construct two bounded words in the
alphabet $\Sigma$ that will play ping-pong on some projective
space and hence will be independent. This will prove Theorem
\ref{thm2}. Since the detailed proof below is somewhat technical
we refer the reader to \cite{note} for an outline of the main
ideas.
Let $\gS\subset\SL_d(\OO_\BK(S))$ be as in the statement of
Theorem \ref{thm2}. Inconsistently with the previous section we
will denote by $\BG$ the Zariski closure of $\langle\gS\rangle$ in
$\SL_d$, and $\gC=\BG (\OO_\BK (S))$. Then $\BG$ is a semisimple
irreducible subgroup in $\SL_d$ and $\gC$ is an $S$--arithmetic
subgroup of $\BG$. We let $G=\prod_{v\in S}\BG(K_v)$ and identify
$\gC$ via the diagonal embedding with the corresponding
$S$--arithmetic lattice in $G$.
In this section, whenever we say that some quantity is a \textit{constant}, we mean that it
may depend only on $d$, $\BK$ and $S$.
The following proposition will allow us to assume that $\gS$ is
finite, hence compact. Let $s\in\BN$ be a constant. We will
specify some condition on $s$ in Paragraph \ref{sbs:step4} (Step
(4)), for the moment we only require it to be at least $r$, the
constant from Proposition \ref{prop:gamma}.
\begin{prop}\label{sigma-two}
There is a constant $f$, such that for
any subset $1\in\gS\subset\gC$ which generates a Zariski dense
subgroup of $\BG$, there is a subset $\gS'$ of $\gS^f$ of
cardinality $\dim (\BG )$ such that:
\begin{enumerate}
\item The Zariski closure $\overline{\langle\gS'\rangle}^Z$ of the group generated by $\gS'$ equals $\BG$, and
\item $(\gS')^s$ consists of semisimple elements.
\end{enumerate}
\end{prop}
Recall the following fact:
\begin{lem}\label{Borel}\label{lem:dominant} \textbf{(see Borel \cite{Bor})} Let $\BG$ be a connected semisimple algebraic
group, and $k\geq 2$ an integer. If $W$ is a non-trivial word in
the free group $F_k$, then the corresponding map $W: \BG^k
\rightarrow \BG$ is dominant.
\end{lem}
\begin{proof}[Proof of Proposition \ref{sigma-two}]
Let $k=\dim (\BG)$, let $W_1,\ldots,W_t$ be all the reduced words
in $F_k$ of length $\leq s$, and consider the map
$w:\BG^k\to\BG^t$ defined by substitution in $(W_1,\ldots,W_t)$.
Let $\Phi\subset\BG$ be a Zariski open subset which consists of
semisimple elements. We shall construct inductively elements
$\gs_i,~i=1,\ldots$ in a bounded power of $\gS$ which $\forall i$
satisfy:
\begin{itemize}
\item There are some $g_{i+1},\ldots,g_k\in\BG$ such that
$w(\gs_1,\ldots,\gs_i,g_{i+1},\ldots,g_k)\in \Phi^t$.
\item $\dim (\overline{\langle\gs_1,\ldots\gs_i\rangle}^Z)\geq i$.
\end{itemize}
In order to construct $\gs_1$ choose some $(g_1,g_2,\ldots,g_k)\in
w^{-1}(\Phi^t)$, which is non-empty by Lemma \ref{lem:dominant} ,
and define
$$
V_1=\{g\in\BG:w(g,g_2,\ldots,g_k)\in\BG^t\setminus\Phi^t\}.
$$
As noted before Lemma \ref{r1}, the Zariski closure $X$ of the
elements in $\gC$ whose projection to one of the factors of $\BG$
is torsion is a proper subvariety of $\BG$. Let $N_1$ be the
constant obtained from Lemma \ref{Bezout} applied to $X\cup V_1$
and take $\gs_1\in\gS^{N_1}\setminus (X\cup V_1)$. It is
straightforward to check that $\chi(V_1)$ (i.e. the sum of the
degrees and dimensions of the irreducible components of $V_1$) can
be bounded independently of the choice of $(g_2,\ldots,g_k)$ and
hence that $N_1$ can be taken to be a constant. Finally, since
$\gs_1$ has infinite order $\overline{\langle\gs_1\rangle}^Z$ has
positive dimension.
To explain the $i$'th step let us suppose that
$\gs_1,\ldots,\gs_{i-1}$ were already constructed. Since
$\gs_1,\ldots,\gs_{i-1}$ are assumed to satisfy the requirements
above, we can chose some new $g_{i+1},\ldots,g_k\in \BG$ for which
the algebraic set
$$
V_i=\{g\in\BG:w(\gs_1\ldots,\gs_{i-1},g,g_{i+1},\ldots,g_k)\in\BG^t\setminus\Phi^t\}
$$
is proper. Additionally, the Zariski connected group
$\BG_i=(\overline{\langle\gs_1,\ldots,\gs_{i-1}\rangle}^Z)^\circ$
cannot be proper normal since by the properties of $\gs_1$ it
projects non-trivially to each simple factor of the semisimple
group $\BG$. If $\BG_i=\BG$ take $\gs_i=1$ and otherwise take $\gd_i\in \gS\setminus N_\BG(\BG_i)$, let $N_i$ be the constant obtained from Lemma \ref{Bezout} applied to $V_i\cup \gd V_i$, chose $\gs_i'\in\gS^{N_i}\setminus
(V_i\cup\gd_i V_i)$, and set $\gs_i=\gs_i'$ if $\gs_i'\notin N_\BG(\BG_i)$ and $\gs_i=\gd_i^{-1}\gs_i'$ otherwise.
Again $N_i$ can be taken to be a constant (independent of the previous choice of $\gs_j,~j<i$, the
choice of $g_j,~j>i$ and the choice of $\gd_i$, since $\chi (V_i\cup\gd_iV_i)$ too can be bounded by a constant). Finally, since $\gs_i$ does not normalize $\BG_i$, $\dim(\overline{\langle\gs_1,\ldots,\gs_i\rangle}^Z>\dim(\overline{\langle\gs_1,\ldots,\gs_{i-1}\rangle}^Z$.
\end{proof}
We will therefore assume that $\gS$ itself is finite and $\gS^s$
consists of semisimple elements (where $s\geq r)$. Applying Proposition
\ref{prop:gamma} we see that up to changing $\gS$ into $\gS^{\gc}$
for some $\gamma\in\Gamma$, we may assume that $\Lambda (A_0)\geq
\|\gS\|$ for some $A_0\in\gS^r$.
We will now fix once and for all a place $v\in S$ for which
$\Lambda_{v}(A_0)=\Lambda (A_0)$. The local field
${\mathbb{K}}_{v}$ has only finitely many extensions of degree at
most $d!$. Let $\tilde{{\mathbb{K}}}_{v}$ be their compositum,
then any semisimple element in $\text{SL}_d({\mathbb{K}}_{v})$ is
diagonalizable in $\text{SL}_d(\tilde{{\mathbb{K}}}_{v})$. Similarly, let $
\tilde{{\mathbb{K}}}$ be the splitting field of $A_0$, and let
$\tilde{S}$ be the set of all places of $\tilde{{\mathbb{K}}}$
extending elements of $S$.
By passing to a suitable wedge power representation $V=\gL^i\BK^d$
for some $i$, $1 \leq i \leq d-1$,
we may assume that $A_0$ has a unique eigenvalue $\alpha_1(A)$ of
maximal $v$--absolute value and that the ratio between
$\alpha_1(A_0)$ and the second largest eigenvalue $\alpha_2(A_0)$
satisfies
\begin{equation*}
\Lambda(A_0)^{d}\geq\Big|
\frac{\alpha_1(A_0)}{\alpha_2(A_0)}\Big|_{v}\geq \Lambda
(A_0)^{\frac{1}{d}} \geq \tau^{1/d},
\end{equation*}
where $\tau$ is the constant introduced in the proof of
Proposition \ref{prop:gamma}. Note that the norm of a matrix in a
wedge power representation such as $V$ is bounded by its original
norm to the power $d$. Thus, we have
\begin{equation} \label{lambda1/lambda2}
|{\alpha_1(A_0)}/{\alpha_2(A_0)}|_{v}^{d^2}\geq \| \gS
\|_{End(V)}.
\end{equation}
We will set $n=dimV$ the dimension of the new representation. Note
that $n \leq 2^d$. Note also that in the canonical basis of the
wedge power space, the matrix elements from $\gS$ (viewed as
matrices in $\SL_n(\BK)$) are still in $\OO_\BK (S)$. Finally
observe that $V$ may not be $\BG$--irreducible. This is not a
fundamental problem. However to keep exposition as simple as
possible we will assume throughout that $V=\BK^{n}_v$ is an
irreducible $\BG$--space with $A_0$ and $\gS$ with matrix
coefficients in $\OO_\BK (S)$ and satisfying the two inequalities
above. At the end we will indicate the changes to be made to
accomodate with the fact that $\gL^i\BK^d$ is not irreducible in
general.
Working with the corresponding projective representation over
$\tilde{{\mathbb{K}}}_{v}$ we will now produce two ping--pong
players in four steps. In the first we will construct a proximal
element, in the second a very contracting one and in the third a
very proximal one. Then we will find a suitable conjugate of the
very proximal element and obtain in this way a second ping--pong
partner.
\subsection{Step 1}
We set $r_0=rd^2$. Let $\{\hat{u}_i\}$ be a basis of
$\tilde{{\mathbb{K}}}_v^{n}$ consisting of
normalized eigenvectors of $A_0$ with corresponding eigenvalues $\{\alpha_i\}$
, such that whenever $\alpha_i=\alpha_j$ the vectors $\hat{u}_i$ and $\hat{u}
_j$ are orthogonal\footnote{In the non-Archimedean case this is
simply taken to mean that $\|\hat u_i-\hat u_j\|=1$.}, and let
$\hat u_i^\perp$ denote the hyperplane spanned by
$\{\hat{u}_j:j\neq i\}$.
\begin{lem}
\label{const:r1} For some constant $r_{1}\in {\mathbb{N}}$,
depending only on $\Gamma $,
\begin{equation*}
d(\hat{u_{i}},\hat{u}_{i}^{\perp })\geq |\frac{\alpha _{1}}{\alpha _{2}}
|_{v}^{-r_{1}}
\end{equation*}
for $i=1,\ldots n$.
\end{lem}
\begin{proof}
First note that since $|\ga_i-\ga_j|_w\leq 2\Lambda (A_0)$ for any
$w\in\ti S$ and $|\ga_i-\ga_j|_w\leq 1$ for any $w\notin \ti S$,
it follows from the product formula that if $\ga_i\neq\ga_j$ then
$$
|\ga_i-\ga_j|\geq (2\Lambda (A_0))^{-|\ti S|}\geq \Lambda
(A_0)^{-|\ti S|(1+\log_{\gt}2)}\geq |\frac{\ga_1}{\ga_2}|_v^{-d|\ti S|(1+\log_{\gt}2)}=
|\frac{\ga_1}{\ga_2}|_v^{-t_0}
$$
where $t_0=d|\ti S|(1+\log_{\gt}2)$. Note also that $|\ti S|\leq
d!|S|$.
Next, observe that it is enough to show that for some constant
$r_1'$,
\begin{equation*}
d(\overrightarrow{u_i},\text{span}\{ \hat{u}_j:\ga_j\neq\ga_i\})\geq |\frac{\ga_1}{\ga_2}|_v^{-r_1'}
\end{equation*}
for any $i$ and any unit vector
$\overrightarrow{u}_i\in\text{span}\{\hat u_j:\ga_j=\ga_i\}$. This
in turn will follow from the next claim which we will prove by
induction on $k$:
\medskip
\noindent {\bf Claim.} For any $k$ there is a positive constant
$t_k$ such that if
$\overrightarrow{u}\in\text{span}\{\hat{u_j}:j\in
I,\ga_j\neq\ga_i\}$ where $I$ is a set of indices with $\dim
(\text{span}\{\hat{u_j}:j\in I,\ga_j\neq\ga_i\})=k$ then
$\|\overrightarrow{u}_i-\overrightarrow{u}\|\geq|\frac{\ga_1}{\ga_2}|_v^{-t_k}$
for any unit vector $\overrightarrow{u}_i\in\text{span}\{\hat
u_j:\ga_j=\ga_i\}$.
\medskip
For $k=1$ we can write $\overrightarrow{u}=\lambda\hat{u}_j$, so
$$
A_0(\overrightarrow{u}_i-\lambda\hat{u}_j)=(\ga_i-\ga_j)\overrightarrow{u}_i+
\ga_j(\overrightarrow{u}_i-\lambda\hat u_j)
$$
i.e.
$$
(A_0-\ga_j)(\overrightarrow{u}_i-\lambda\hat{u}_j)=
(\ga_i-\ga_j)\overrightarrow{u}_i,
$$
which implies that (recall $r_0=rd^2$)
$$
\|\overrightarrow{u}_i-\lambda\hat
u_j\|_v\geq\frac{|\ga_i-\ga_j|_v}{\|A_0\|_v+|\ga_j|_v}\geq
|\frac{\ga_1}{\ga_2}|_v^{-t_0-(r_0+d\log_{\gt}2)}:=
|\frac{\ga_1}{\ga_2}|_v^{-t_1}.
$$
Now suppose $k>1$. We can write
$\overrightarrow{u}=\sum\lambda_j\overrightarrow{u}_j$ where the
$\overrightarrow{u}_j$'s are normalized eigenvectors of different
eigenvalues. Abusing indices, we will assume that
$\overrightarrow{u}_j$ corresponds to the eigenvalue $\ga_j$. Now
$$
A_0(\overrightarrow{u}_i-\sum\lambda_j\overrightarrow{u}_j)=
\ga_i(\overrightarrow{u}_i-\sum\lambda_j\overrightarrow{u}_j)+
\sum_j(\ga_i-\ga_j)\lambda_j\overrightarrow{u}_j,
$$
therefore
$$
(A_0-\ga_i)(\overrightarrow{u}_i-\sum\lambda_j\overrightarrow{u}_j)=
\sum_j(\ga_i-\ga_j)\lambda_j\overrightarrow{u}_j.
$$
Note that we may assume that $\|\overrightarrow{u}\|_v\geq 1/2$,
for otherwise the statement is obvious, and hence for some $j_0$,
$|\lambda_{j_0}|_v\geq 1/(2n)$ and by the induction hypothesis
$$
\|\sum_j(\ga_i-\ga_j)\lambda_j\overrightarrow{u}_j\|\geq
|\lambda_{j_0}|_v|\frac{\ga_1}{\ga_2}|_v^{-t_0-t_{k-1}}\geq
\frac{1}{2n}|\frac{\ga_1}{\ga_2}|_v^{-t_0-t_{k-1}}.
$$
It follows that
$$
\|\overrightarrow{u}_i-\sum\lambda_j\overrightarrow{u}_j\|_v\geq
\frac{1}{2n}|\frac{\ga_1}{\ga_2}|_v^{-t_0-t_{k-1}}\frac{1}{\|A_0\|_v+|\ga_i|_v}\geq
|\frac{\ga_1}{\ga_2}|_v^{d\log_{\gt}\frac{1}{2n}-t_0-t_{k-1}
-(r_0+d\log_{\gt}2)}:=|\frac{\ga_1}{\ga_2}|_v^{-t_k}.
$$
\end{proof}
As a consequence we obtain that for some constant $r_2$, depending
only on $\Gamma$, which we may take $\geq r_1$, we have:
\begin{cor}
\label{D} There is a matrix $D\in
\text{SL}_{n}(\tilde{{\mathbb{K}}}_{v})$ such that:
\begin{itemize}
\item $\Vert D\Vert^2 ,\Vert D^{-1}\Vert^2 \leq |\frac{\alpha _{1}}{\alpha _{2}}
|_{v}^{r_{2}}$, and
\item $A_{0}^{D}=DA_0D^{-1}$ is diagonal.
\end{itemize}
\end{cor}
\begin{proof}
Let $D$ be the matrix defined by the condition $D(\hat
u_i)=e_i,~i=1,\ldots,n$. Clearly $|\text{det}(D^{-1})|_v\leq 1$.
Since $D^{-1}=\text{det}(D^{-1})\text{Adj}(D)$ and since
$\|\text{Adj}(D)\|\leq n! \| D\|^{n-1}$ it is enough to prove that
$\| D\|\leq |\frac{\ga_1}{\ga_2}|_v^{r_2'}$.
Let $\hat{u}$ be a unit vector, and write $\hat u=\sum\lambda_i
e_i$. Then for some $i_0$ we have $|\lambda_{i_0}|_v\geq 1/n$.
Since $D^{-1}(\hat u)=\sum\lambda_i\hat{u}_i$, it follows from the
previous lemma that
$$
\| D^{-1}(\hat u)\| =\|\sum\lambda_i\hat{u}_i\|=
\|\lambda_{i_0}\hat{u}_{i_0}+\sum_{j\neq i_0}\lambda_j\hat u_j\|\geq
\frac{1}{n}|\frac{\ga_1}{\ga_2}|_v^{-r_1}\geq
|\frac{\ga_1}{\ga_2}|_v^{-r_2'},
$$
i.e. $\| D\|\leq |\frac{\ga_1}{\ga_2}|_v^{r_2'}.$
\end{proof}
We derive the following proposition and thus conclude the first
step in our construction of ping--pong players:
\begin{prop}[The proximal element $A_1$]
Whenever $r_{3}\geq 8r_{2}$, the element $A_{1}=A_{0}^{r_{3}}$ is $(|
\frac{\alpha _{1}}{\alpha _{2}}|_{v}^{-r_{1}},|\frac{\alpha _{1}}{\alpha _{2}
}|_{v}^{-(r_{3}/2-2r_{2})})$--proximal with attracting point
$[\hat{u}_{1}]$
and repelling hyperplane $[\hat{u}_{1}^{\perp }]=[{\text{span}(\hat{u}
_{2},\ldots ,\hat{u}_{n})}]$.
\end{prop}
\begin{proof}
The diagonal matrix $DA_0^{r_3}D^{-1}$ is obviously
$|\frac{\ga_1}{\ga_2}|_v^{-r_3/2}$--contracting with attracting
point $[{e}_1]$ and repelling hyperplane $[{\text{span}(e_2,\ldots
,e_n)}]$. Since $\| D\|,\| D^{-1}\|\leq
|\frac{\ga_1}{\ga_2}|_v^{r_2}$, $D$ is
$|\frac{\ga_1}{\ga_2}|_v^{2r_2}$ bi-Lipschitz. It follows that
$A_0^{r_3}$ is
$|\frac{\ga_1}{\ga_2}|_v^{-(r_3/2-2r_2)}$--contracting. Finally,
Lemma \ref{const:r1} implies that $d([\hat{u}_1],[\hat
u_1^{\perp}])\geq |\frac{\ga_1}{\ga_2}|_v^{-r_1}$
\end{proof}
\subsection{Step 2}
Our next goal is to build a very contracting element out of the
matrix $A_1$. To achieve this, we will find some bounded
word $B_1$ in $\gS$ which will be in ``general position"
with respect to $A_1$. Then $A_2=A_1^{r_7}B_1A_1^{-r_7}$ will be
our candidate. In this process we will ``lose" the information we
have on the position of the repelling neighborhoods. However we
will still have a good control on the positions of the attracting
points of $A_2$ and $A_2^{-1}$, a control which will turn crucial
in the following step when producing a very proximal element
$A_3$. The key idea is that while $B_1$ sends the eigen-directions
of $A_1$ away from the eigen-hyperplanes of $A_1$, we can estimate
this quantitatively by giving an explicit lower bound. In order to
formulate a precise statement, we will need to introduce another
basis of eigenvectors for $A_1$.
\begin{lem}
\label{nice-vectors} For each $k\leq n$ there is an eigenvector $
\overrightarrow{u}_{k}\in \overline{{\mathbb{K}}}^{n}$ for $A_{0}$
with corresponding eigenvalue $\alpha _{k}$ whose coordinates are
$\ti S$--integers and whose $w$--norm is at most $|\alpha
_{1}/\alpha _{2}|_{v}^{r_{4}}$ for any $w\in \ti S$, where $r_{4}$
is some constant depending only on $r_{0},d$ and the size of $S$.
\end{lem}
\begin{proof}
Recall from inequality (\ref{lambda1/lambda2}) that for each $w\in
S$ we have $\| A_0\|_w\leq |\ga_1/\ga_2|_v^{r_0}$ (where
$r_0=rd^2$). Suppose that $\ga_i$ has multiplicity $k$, say
$\ga_i=\ga_{i+1}=\ldots =\ga_{i+k-1}$, then we can pick $k$
indices between $1$ and $n$ such that the $(n-k)\times (n-k)$
matrix obtained by restricting $A_0 -\ga_i$ to the remaining
indices is invertible. We can then define
$\overrightarrow{u_{i+j}},~j\leq k-1$ to be the eigenvector of
$\ga_{i+j}=\ga_i$ whose entries corresponding to the chosen $k$
indices are all $0$ except the $(j-1)$'th one which equals the
determinant of the $(n-k)\times (n-k)$ submatrix. Solving the
corresponding linear equation, it is easy to verify that these
vectors satisfy the requirement with respect to some bounded
constant $r_4$.
\end{proof}
In analogy to our previous notations, we will denote by $\overrightarrow{u_i
}^\bot$ the span of the $\overrightarrow{u_j}$'s, $j\neq i$. Note
that since $\alpha_1$ has multiplicity one, we have $[\hat
u_1]=[\overrightarrow{u_1}]$ and $[\hat
u_1^\bot]=[\overrightarrow{u_1}^\bot]$.
\begin{defn}\label{defn:general-position}
Let $N$ be an integer and $v_1,\ldots,v_n\in\overline{\BK}^n$ a
basis. We will say that a matrix $C\in\SL_n(\overline{\BK} )$ is
in $N$--{\it general position} with respect to $\{
v_1,\ldots,v_n\}$ if
\begin{itemize}
\item for any $1\leq i,j\leq n$, not necessarily distinct, both
vectors $Cv_i$ and $C^{-1}v_i$ do not lie in the hyperplane
spanned by $\{v_k\}_{k\neq j}$, and
\item for any $n$ integers $1\leq i_1<\ldots <i_n\leq N$ and any
$1\leq j\leq n$ the vectors $C^{i_1}v_j,\ldots,C^{i_n}v_j$ are
linearly independent.
\end{itemize}
\end{defn}
For a fixed $N$, the varieties
$$
X(N,v_1,\ldots,v_n)=\{g\in\SL_n(\overline{\BK} ):g~\text{is {\it not} in $N$--general position w.r.t.}~\{v_i\}_{i=1}^n\}
$$
are all conjugate inside $\SL_n(\overline{\BK} )$.
Since $\BG$ is Zariski connected and irreducible, one can derive
that $X(N,v_1,\ldots,v_n)\cap\BG$ is a proper subvariety of $\BG$.
Hence by Lemma \ref{Bezout} for any $N$ there is a constant
$m_2(N)$ such that for any set $\Omega$ which generates a Zariski
dense subgroup of $\BG$, and any basis $\{ v_i\}_{i=1}^n$ of
$\BK^n$, there is an element in $\Omega^{m_2(N)}$ which is in
$N$--general position with respect to $\{v_i\}_{i=1}^n$. In
particular we may find $B_1\in \gS^{m_2}$ (with $m_2=m_2(2n-1)$)
which is in $(2n-1)$--general position with respect to
$\{\overrightarrow{u}_i\}_{i=1}^n$.
In the proof of Proposition \ref{very-contracting} we will make
use of the following lemma only for $i=n$ and $j=1$.
\begin{lem}
\label{r7}\label{const:r5} For some positive bounded constant
$r_{5}$ we have
\begin{equation*}
d((B_1^{\pm 1})\cdot [\overrightarrow{u_{i}}],[\overrightarrow{u_{j}^{\perp }}
])>|\frac{\alpha _{1}}{\alpha _{2}}|_{v}^{-r_{5}},
\end{equation*}
for any $i,j\leq n$.
\end{lem}
\begin{proof}
For each $w\in\ti S$, the $w$--absolute values of the coordinates
of $B_1(\overrightarrow{u}_i)$ are at most
$|\ga_1/\ga_2|_v^{m_2r_0+r_4}$. Consider the determinant
$$
\mathcal{D}_{\pm 1}=\text{det}(B^{\pm 1}(\overrightarrow{u}_i),\overrightarrow{u}_1,\ldots,
\overrightarrow{u}_{j-1},\overrightarrow{u}_{j+1},\ldots,\overrightarrow{u}_n).
$$
This is again an $\ti S$--integer and its $w$--absolute value is
at most $|\ga_1/\ga_2|_v^{m_2r_0+nr_4}$. Since $B_1$ is in general
position with respect to $\{\overrightarrow{u}_i\}_{i=1}^n$ we
have $\mathcal{D}_{\pm 1}\neq 0$. By the product formula
$\prod_{\text{all places}}|\mathcal{D}_{\pm 1}|_w=1$ and hence
$\prod_{w\in\ti S}|\mathcal{D}_{\pm 1}|_w\geq 1$. It follows that
$$
|\mathcal{D}_{\pm 1}|_v\geq |\ga_1/\ga_2|_v^{-(m_2r_0+nr_4)|\ti S|}.
$$
Now since all the vectors involved in this determinant have
$v$--norm at most $|\ga_1/\ga_2|_v^{m_2r_0+r_4}$, the distance
between each of them to the hyperplane spanned by the others is at
least
$$
\frac{|\mathcal{D}_{\pm 1}|_v}{|\ga_1/\ga_2|_v^{(m_2r_0+r_4)(n-1)}}\geq
|\ga_1/\ga_2|_v^{-(m_2r_0+nr_4)(|\ti S|+n-1)}.
$$
The lemma follows.
\end{proof}
We will also need the following:
\begin{lem}
There exists some $\epsilon =\epsilon (n)$, such that if $d=\text{diag}
(d_{1},\dots ,d_{n})\in \text{SL}_{n}(\tilde{{\mathbb{K}}}_{v})$
is a
diagonal matrix with $d_{1}\geq d_{2}\geq \ldots \geq d_{n}$, then $[d]$ is $
2$--Lipschitz on the $\epsilon$--ball around $[e_{1}]$.
\end{lem}
\begin{proof}
The lemma follows by a direct simple computation. In the
non-Archimedean case a diagonal matrix is $1$--Lipschitz on the
open unit ball around $[e_1]$. In the Archimedean case the same is
true for the metric which is induced on $\BP (\ti\BK_v^n)$ from
the $L^{\infty}$ norm on $\ti\BK_v^n$. Since the renormalization
map from the euclidean unit sphere to the $L^{\infty}$ unit sphere
is $C^1$ around $e_1$ with differential $1$ at $e_1$ it has a
bi-Lipschitz constant arbitrarily close to $1$ in a small
neighborhood of $e_1$. The result follows.
\end{proof}
We are now able to formulate:
\begin{prop}[The very contracting element $A_2$]
\label{very-contracting} For any $r_{6}\in\BN$, there exists
$r_{7}\in\BN$ such that the element
$A_{2}=A_{1}^{r_{7}}B_1A_{1}^{-r_{7}}$ is $|\frac{\alpha
_{1}}{\alpha _{2}}|_{v}^{-r_{6}}$ very contracting, with both
attracting points (of the
element and its inverse) lying in the $|\frac{\alpha _{1}}{\alpha _{2}}
|_{v}^{-r_{6}}$ ball around $[\hat{u}_{1}]$.
\end{prop}
The proof of Proposition \ref{very-contracting} relies on
Proposition \ref{contracting-properties}, as well as the last two
lemmas:
\begin{proof}[Proof of Proposition \ref{very-contracting}]
Let $r_7\in\BN$ be arbitrary, to be determined later. By the
previous lemma, the diagonal matrix $DA_1^{-r_7}D^{-1}$ is
$2$--Lipschitz on the on the $\gep (n)$--ball around
$[e_n]=D[\hat{u}_n]$. By Corollary \ref{D} $\| D^{\pm 1}\|^2\leq
|\ga_1/\ga_2|_v^{r_2}$ which implies that $D^{\pm 1}$ are
$|\ga_1/\ga_2|_v^{2r_2}$ Lipschitz (on the entire projective
space, see Section \ref{prelim} (iv)). It follows that
$A_1^{-r_7}$ is $2|\ga_1/\ga_2|_v^{4r_2}$ Lipschitz on the
$\gep\cdot |\ga_1/\ga_2|_v^{-2r_2}$ ball around $[\hat u_n]$, or
in other words, that $A_1^{-r_7}$ is
$|\ga_1/\ga_2|_v^{d\log_{\gt}2+4r_2}$ Lipschitz on the
$|\ga_1/\ga_2|_v^{d\log_{\gt}\gep -2r_2}$ ball around $[\hat
u_n]$.
Now since $\| B_1^{\pm 1}\|_v\leq |\ga_1/\ga_2|^{m_2r_0}$, the
matrices $B_1^{\pm 1}$ are $|\ga_1/\ga_2|^{2m_2r_0}$ Lipschitz on
the projective space, and hence the matrices $B_1^{\pm
1}A_1^{-r_7}$ are $|\ga_1/\ga_2|_v^{d\log_{\gt}2+4r_2+2m_2r_0}$
Lipschitz on the $|\ga_1/\ga_2|_v^{d\log_{\gt}\gep -2r_2}$ ball
around $[\hat u_n]$.
Take
$$
c^*=\max\{ 2r_2-d\log_{\gt}\gep,~~d\log_{\gt}2+4r_2+2m_2r_0+2r_7\},
$$
then the $|\ga_1/\ga_2|_v^{-c^*}$--ball $\gO$ around $[\hat u_n]$
is mapped under $B_1A_1^{-r_7}$ (resp. under $B_1^{-1}A_1^{-r_7}$)
into the $|\ga_1/\ga_2|_v^{-2r_7}$--ball around $B_1[\hat u_n]$
(resp. around $B_1^{-1}[\hat u_n]$). By Lemma \ref{const:r5}
$$
d(B_1^{\pm 1}[\hat u_n],[\hat u_1^\perp])\geq
|\ga_1/\ga_2|_v^{-r_5}.
$$
Note that without loss of generality we can set $[\hat u_n]$ to be
equal to $[\overrightarrow{u_n}]$. Also we may assume that
$|\ga_1/\ga_2|_v^{-r_5}<1/\sqrt{2}$ and that $r_7\geq r_5$.
Therefore, $B_1A_1^{-r_7} \gO$ and $B^{-1}_1A_1^{-r_7} \gO$ lie
outside the $|\ga_1/\ga_2|_v^{-2r_5}$ neighborhood of $[\hat
u_1^\bot]$. It follows that both sets $DB_1^{\pm 1}A_1^{-r_7}\gO$
lie outside the $|\ga_1/\ga_2|_v^{-r_5-2r_2}$ neighborhood of
$D[\hat u_1^\bot ]=[\text{span}\{e_2,\ldots, e_n\} ]$. By
Proposition \ref{contracting-properties} $(1)$ applied to the
diagonal matrix $DA_1^{r_7}D^{-1}$, it is
$|\ga_1/\ga_2|_v^{-r_7+2(r_5+2r_2)}$--Lipschitz outside the
$|\ga_1/\ga_2|_v^{-r_5-2r_2}$ neighborhood of
$[\text{span}\{e_2,\ldots, e_n\} ]$, and hence $A_1^{r_7}D^{-1}$
is $|\ga_1/\ga_2|_v^{-r_7+2(r_5+3r_2)}$-Lipschitz there. Thus
$A_1^{r_7}B_1^{\pm 1}A_1^{-r_7}=(A_1^{r_7}D^{-1})D(B_1^{\pm
1}A_1^{-r_7})$ are both
$$
|\ga_1/\ga_2|_v^{-r_7+c^{**}}-\text{Lipschitz}
$$
on $\gO$, where we have set
$$
c^{**}=2(r_5+3r_2)+(d\log_{\gt}2+4r_2+2m_2r_0)+2r_2.
$$
It follows from parts (2) and (3) of Proposition
\ref{contracting-properties} that the elements
$A_1^{r_7}B_1^{\pm}A_1^{-r_7}$ are both
$|\ga_1/\ga_2|_v^{\frac{1}{2}[-r_7+c^{**}]}$ contracting. Thus
taking
$$
r_7\geq 2r_6+c^{**}
$$
we guarantee that $A_1^{r_7}B_1A_1^{-r_7}$ is
$|\ga_1/\ga_2|_v^{-r_6}$ very contracting.
Now suppose further that
$$
r_7\geq 2\max\{2r_6,c^*\}+c^{**},
$$
then our elements $A_1^{r_7}B_1^{\pm 1}A_1^{-r_7}$ are
$|\ga_1/\ga_2|_v^{-\max\{2r_6,c^*\}}$--very contracting. Moreover
$\gO$ is a $|\ga_1/\ga_2|_v^{-c^*}$--ball, hence contains a point
$p^+$ (resp. a point $p^-$) that is at least
$|\ga_1/\ga_2|_v^{-c^*}$-away from the repelling hyperplane of
$A_1^{r_7}B_1A_1^{-r_7}$ (resp. of
$A_1^{r_7}B_1^{-1}A_1^{-r_7}$). It follows that
$A_1^{r_7}B_1^{\pm}A_1^{-r_7}$ maps the points $p^\pm$
respectively into the $|\ga_1/\ga_2|_v^{-2r_6}$--ball around the
corresponding attracting points $t^\pm$ of $A_1^{r_7}B_1^{\pm
1}A_1^{-r_7}$, i.e.
$$
d(A_1^{r_7}B_1^{\pm 1}A_1^{-r_7}
(p^\pm),t^\pm)\leq |\ga_1/\ga_2|_v^{-2r_6}.
$$
Additionally the element $A_1^{r_7}$ is
$|\ga_1/\ga_2|_v^{-r_7/2+4r_2}$--contracting with attracting point
$[\hat u_1]$ and repelling hyperplane $[\hat u_1^\bot]$, and since
the point $B_1[\hat u_n]$ lies outside the
$|\ga_1/\ga_2|_v^{-r_5}$ neighborhood of $[\hat u_1^\bot]$,
assuming further that $r_7\geq 2r_5+8r_2$, we get that this point
is mapped under $A_1^{r_7}$ to the
$|\ga_1/\ga_2|_v^{-r_7/2+4r_2}$--ball around $[\hat u_1]$. We
conclude that $[\hat u_n] \in \gO$ is mapped under
$A_1^{r_7}B_1A_1^{-r_7}$ into the
$|\ga_1/\ga_2|_v^{-r_7/2+4r_2}$--ball around $[\hat u_1]$.
Finally since $A_1^{r_7}B_1A_1^{-r_7}$ is
$|\ga_1/\ga_2|_v^{-r_7+c^{**}}$ Lipschitz on $\gO$, we get that
\begin{eqnarray*}
d(t^+,[\hat u_1])
&\leq& d(t^+,A_1^{r_7}B_1A_1^{-r_7}
p^+)+\text{diam}(A_1^{r_7}B_1A_1^{-r_7}\gO) +d(A_1^{r_7}B_1A_1^{-r_7}[\hat
u_n],[\hat u_1])\\
&\leq& |\ga_1/\ga_2|_v^{-2r_6} +2|\ga_1/\ga_2|_v^{-r_7+c^{**}-c^*}
+|\ga_1/\ga_2|_v^{-r_7/2+4r_2}.
\end{eqnarray*}
By choosing $r_7$ sufficiently large, we can make the last
quantity smaller the $|\ga_1/\ga_2|_v^{-r_6}$, that is
$$
d(t^+,[\hat u_1])\leq |\ga_1/\ga_2|_v^{-r_6}.
$$
The same computation with $t^-,p^-$ replacing $t^+,p^+$ gives
$d(t^-,[\hat u_1])\leq |\ga_1/\ga_2|_v^{-r_6}$. This finishes the
proof of the proposition.
\end{proof}
\subsection{Step 3}
Our next step is to use $A_2=A_1^{r_7}B_1A_1^{-r_7}$ to build a
very proximal element. Note that we haven't specified any
condition on the constants $r_6,r_7$ from Lemma \ref
{very-contracting} yet. We will show that for some suitable $k\leq
2n-1$ the matrix $B_1^kA_2$ is very proximal.
Let $\overrightarrow{u}_1\in\tilde{{\mathbb{K}}}_v\cdot\hat{u}_1$
be an eigenvector of $A_1$ corresponding to $\alpha_1$ as in
Lemma \ref{nice-vectors}, i.e. the coordinates of $\overrightarrow{u}_1$ are $
\tilde{S}$-integers, and $|\overrightarrow{u}_1|_w\leq
|\alpha_1/\alpha_2|_v^{r_4}$, for any $w\in\ti S$.
For any $k\leq 2n-1$ we have
\begin{equation*}
\| B_1^k(\overrightarrow{u}_1)\|_w\leq \| B_1\|_w^{2n-1}\|\overrightarrow{u}
_1\|_w\leq |\alpha_1/\alpha_2|_v^{(2n-1)m_2r_0+r_4},
\end{equation*}
for any $w\in\ti S$, while for any $w\notin\ti S$ the $w$-norm of this vector is $
\leq 1$. It follows that for any $1\leq k_1<\ldots<k_n\leq 2n-1$
we have
\begin{equation*}
|\text{det}(B_1^{k_1}(\overrightarrow{u}_1),\ldots,B_1^{k_n}(\overrightarrow{u}
_1))|_w \leq |\alpha_1/\alpha_2|_v^{((2n-1)m_2r_0+r_4)n}
\end{equation*}
for any $w\in\ti S$, and
\begin{equation*}
|\text{det}(B^{k_1}(\overrightarrow{u}_1),\ldots,B^{k_n}(\overrightarrow{u}
_1)|_w \leq 1
\end{equation*}
for any $w\notin\ti S$. Since $B_1$ is in $(2n-1)$--general
position with respect to the $\{\overrightarrow{u}_i\}$'s, this
determinant is not zero, and hence by the product formula
\begin{equation*}
\prod_{\text{{\tiny over all places}}} |\text{det}(B_1^{k_1}(
\overrightarrow{u}_1),\ldots,B_1^{k_n}(\overrightarrow{u}_1)|_w
=1,
\end{equation*}
which implies that
\begin{equation*}
\prod_{w\in\ti S} |\text{det}(B_1^{k_1}(\overrightarrow{u}_1),\ldots,B_1^{k_n}(
\overrightarrow{u}_1))|_w \geq 1.
\end{equation*}
We conclude:
\begin{cor}
For any $w\in\ti S$, and in particular for $w=v$
\begin{equation*}
|\text{det}(B_1^{k_{1}}(\overrightarrow{u}_{1}),\ldots ,B_1^{k_{n}}(
\overrightarrow{u}_{1}))|_{w}\geq |\alpha _{1}/\alpha
_{2}|_{v}^{-((2n-1)m_2r_{0}+r_{4})n|\ti S|}.
\end{equation*}
\end{cor}
We will need also the following:
\begin{lem}
\label{v-f-H} Suppose that $\overrightarrow{v}_{1},\ldots ,\overrightarrow{v}
_{n}$ are any $n$ vectors in $\tilde{{\mathbb{K}}}_{v}^{n}$
satisfying
\begin{itemize}
\item $\Vert \overrightarrow{v}_{i}\Vert _{v}\leq t^{c^{\prime
}},~\forall i\leq n$, and
\item $|\text{det}(\overrightarrow{v}_{1},\ldots ,\overrightarrow{v}
_{n})|_{v}\geq t^{-c^{\prime \prime }}$,
\end{itemize}
for some $c^{\prime },c^{\prime \prime }\in {\mathbb{N}}$ and
$t>0$.
Then for any hyperplane $H\subset \tilde{\BK}^{n}_v$ there is
$i\leq n
$ such that $d([\overrightarrow{v}_{i}],[H])\geq \frac{1}{\lambda_{1}\lambda _{n-1}}
t^{-c^{\prime \prime }-(n-1)c^{\prime }}$ in the $\tilde{\BK}_v$
projective space, where $\lambda_{k}$ is the volume of the
$k$--dimensional unit ball (in particular $\lambda _{k}=1$ in the
non-Archimedean case).
\end{lem}
\begin{proof}
Let $f$ be a linear form such that $\|f\|=1$ and $H=\ker(f)$, then
the volume of $\{x\in\tilde{\BK}_v^n : |f(x)|_v \leq |a|_v,
\|x\|_v \leq |b|_v\}$ is bounded above by
$\lambda_{1}\lambda_{n-1}|a|_v|b|_v^{n-1}$ -- the volume of a
``cylinder'' with base radius $|b|_v$ and ``height'' $2|a|_v$, for
any $a,b \in \tilde{\BK}_v$. Since
$d([x],[H])=\frac{|f(x)|_v}{\|x\|_v}$, we get the desired
conclusion by comparing this volume to the determinant of the
${\overrightarrow{v}_{i}}$'s.
\end{proof}
Setting $c^{\prime}=(2n-1)m_2r_0+r_4$ and
$c^{\prime\prime}=((2n-1)m_2r_0+r_4)n|\ti S|$ we
get some constant\footnote{
Note that we can fix $r_{8}$ before determining $r_6,r_7$.} $r_8$,
such that whenever
$\overrightarrow{v}_1,\ldots,\overrightarrow{v}_n$ are as in Lemma
\ref{v-f-H} with $t=|\alpha_1/\alpha_2|_v$ and $[H]$ is some
projective
hyperplane, there is one $[\overrightarrow{v}_i]$ at distance at least $
|\alpha_1/\alpha_2|^{-r_8}_v$ from $[H]$, in particular:
\begin{lem}
\label{r10}\label{const:r8} For any $1\leq k_{1}<k_{2}<\ldots
<k_{n}\leq 2n-1 $ and any hyperplane $H\subset \tilde{\BK}^{n}_v$
there exists $i\leq n $ such that
\begin{equation*}
d([B_1^{\pm k_{i}}\hat{u}_{1}],[H])\geq |\alpha _{1}/\alpha
_{2}|_{v}^{-r_{8}}.
\end{equation*}
\end{lem}
By the pigeonhole principle, we conclude:
\begin{cor}\label{pigeon}
\label{2n-1} For any two hyperplanes $H_{1},H_{2}\subset
\tilde{\BK}^{n}_v$ there is some $k\leq 2n-1$ such that we have
simultaneously
\begin{equation*}
d([B_1^{k}\hat{u}_{1}],[H_{1}])\geq |\alpha _{1}/\alpha
_{2}|_{v}^{-r_{8}}, d([B_1^{-k}\hat{u}_{1}],[H_{2}])\geq |\alpha
_{1}/\alpha _{2}|_{v}^{-r_{8}}.
\end{equation*}
\end{cor}
Now let $[H^+],[H^-]$ be the repelling hyperplanes for the $
|\alpha_1/\alpha_2|_v^{-r_6}$--very contracting element $
A_2=A_1^{r_7}B_1A_1^{-r_7}$ and its inverse, and take the
corresponding $k$ in Corollary \ref {2n-1}. Recall that the
attracting points $t^+,t^-$ of $A_2^{\pm 1}$ are both at distance
at most $|\alpha_1/\alpha_2|_v^{-r_6}$ from $[\hat u_1]$. We thus
obtain:
\begin{prop}[The very proximal element $X$]\label{prop:very-proximal}
Assume that $r_{8}>2(2n-1)r_{0}$, then the element
$X=B_1^{k}A_{2}$ is $(\rho,\gd)$--very proximal with
$$
\rho=|\alpha_{1}/\alpha _{2}|_{v}^{-2m_2kr_{0}}(|\alpha _{1}/
\alpha_{2}|_{v}^{-r_{8}}-|\alpha _{1}/
\alpha_{2}|_{v}^{-r_{6}+4m_2kr_{0}}),\text{~and~}
\gd=|\alpha_{1}/\alpha _{2}|_{v}^{-r_{6}+4m_2kr_{0}}
$$
and with repelling hyperplanes
\begin{equation*}
\lbrack H_{X}^{+}]=[H^{+}],~~[H_{X}^{-}]=B_1^{k}[{H}^{-}]
\end{equation*}
and attracting points
\begin{equation*}
\lbrack t_{X}^{+}]=B_1^{k}t^{+},~~[t_{X}^{-}]=t^{-}.
\end{equation*}
\end{prop}
\begin{proof}
Since $\|B_1^{\pm 1}\|_v\leq |\ga_1/\ga_2|_v^{m_2r_0}$, $B_1$ is
$|\ga_1/\ga_2|_v^{4m_2r_0}$ bi-Lipschitz on the entire projective
space. This implies that $X=B_1^kA_2$ is
$|\ga_1/\ga_2|_v^{-r_6+4m_2kr_0}$ very contracting with the
specified attracting points and repelling hyperplanes, and that
\begin{eqnarray*}
d(B_1^k (t^+),[H^+])\geq
d(B_1^k[\hat u_1],[H^+])-d(B_1^k [\hat
u_1],B_1^k (t^+))\geq
|\ga_1/\ga_2|_v^{-r_8}-|\ga_1/\ga_2|_v^{-r_6+4km_2r_0},
\end{eqnarray*}
and
\begin{eqnarray*}
d(t^{-},B_1^{k}[H^{-}])\geq\|B_1^{\pm k}\|_v^{-4}d(B_1^{-k}
(t^{-}),[H^{-}])
\geq
|\ga_1/\ga_2|_v^{-4m_2kr_0}(|\ga_1/\ga_2|_v^{-r_8}-|\ga_1/\ga_2|_v^{-r_6+4km_2r_0})
\end{eqnarray*}
\end{proof}
Taking $r_6>>r_8$ sufficiently large (after choosing $r_8$
sufficiently large) we may assume that:
\begin{equation*}
\rho=|\alpha_1/\alpha_2|_v^{-r_8-2m_2kr_0}(1-|\alpha_1/\alpha_2|_v^{r_8-r_6+6m_2kr_0})
\geq\frac{1}{2}|\alpha_1/\alpha_2|_v^{-2r_8}.
\end{equation*}
Set
\begin{equation*}
r_9=2r_8+d\log_\gt 2,~~r_{10}=r_6-4m_2kr_0,
\end{equation*}
Then we get that $X=B_1^kA_1^{r_7}B_1A_1^{-r_7}$ is $(|\alpha_1/
\alpha_2|_v^{-r_9},|\alpha_1/\alpha_2|_v^{-r_{10}})$--very
proximal. The matrix $X$ is our first ping--pong player.
\subsection{Step 4}\label{sbs:step4}
The last step of the proof consists in finding a second ping--pong
partner $Y$ by conjugating $X$ by a suitable bounded word in the alphabet $\gS$. This is performed in quite the same way as in Step 3, so
we only sketch the proof here.
Note first that $X$ is a word in $\gS$ of length at most
$2(2n-1)m_2+2r_7r_3r$. Therefore, by
requiring $s$ from Proposition \ref{sigma-two} to be at least this constant, we can
assume that $X$ is semisimple. Let $[\hat v_1]$ (resp. $[\hat
v_n]$) be the eigendirection of the maximal (resp. minimal)
eigenvalue of $X$.
Let $B_2$ be a word in $\gS$ which is in $(2n-1)^2$--general
position with respect to the eigenvectors of $X$ (chosen as in
Lemma \ref{nice-vectors}). Again by Lemma \ref{Bezout} and the
discussion following Definition \ref{defn:general-position}, we
may find $B_2$ as a word of length $\leq m_2((2n-1)^2)$.
We can then apply the same pigeonhole argument as in Corollary
\ref{pigeon} and obtain an index $k'\leq (2n-1)^2$ such that
$B_2^{k'}[\hat v_1]$ and $B_2^{k'}[\hat v_n]$ are both far away
from the repelling hyperplanes $[H_X^+]$ of $X$ and $[H_X^-]$ of
$X^{-1}$ (i.e. $|\alpha_1/\alpha_2|_v^{-r_{11}}$--apart for some
other constant $r_{11}$).
Setting $Y=B_2^{k'}XB_2^{-k'}$, we see that some bounded power
$Y^{r_{12}}$ of $Y$ is very proximal with attracting and repelling
points $B_2^{k'}[\hat v_1]$ and $B_2^{k'}[\hat v_n]$ and repelling
hyperplanes $B_2^{k'}[H_X^+]$ and $B_2^{k'}[H_X^-]$. Since those
points are away from the repelling hyperplanes of $X$ (or any
power of $X$), we conclude that, after taking a larger power
$r_{13}\geq r_{12}$ if necessary, $Y^t$ and $X^{t}$
play ping--pong, and hence independent, for any $t\geq r_{13}$.
\begin{rem}
As mentioned at the beginning of Section \ref{sec:ping-pong} we
assumed throughout that the representation space $V$ was
irreducible. Lemma \ref{nice-vectors} as well as the rest of the
argument above, relies on the assumption that the entries of the
elements of $\gS$ viewed as matrices acting on $V$ are
$S$--arithmetic. However, in general our wedge representation $V$
might be reducible, and we have to replace it with some
irreducible subquotient where this assumption may not hold. In
order to cope with this problem we argue as follows. We change the
representation space from $V$ to an irreducible subquotient
$V_0/W$ where $V_0,W$ are invariant subspaces of $V$. One can
carry out the proof of Lemma \ref{nice-vectors} in $V$ and first
treat the eigenvectors in $W$, then those in $V_0\setminus W$ and
finally take the projections of those to $V_0/W$. This would yield
an analogous statement for $V_0/W$ which is sufficient for the
whole argument. Note also that in characteristic zero, as
$\BG$ is semisimple, $V$ is completely reducible so our
irreducible representation is a sub-representation of the wedge
power, rather than a subquotient, and hence, in this case, we may take $V_0$
instead of $V$ without further changes.
\end{rem}
This completes the proof of Theorem \ref{thm2}. \qed
\medskip
\section{A Zariski dense free subgroup in characteristic zero}\label{sec:Z-dense}
We will now give two stronger versions of Theorem
\ref{thm1} which are useful for applications. Since all the applications we have in mind are for fields of characteristic zero, we allow ourselves to make this restriction, although we believe that it is unnecessary.
\begin{thm}\label{thm:Z-dense}
Let $K$ be a field of characteristic zero, $\BH$ a Zariski
connected semisimple $K$--group and $\gC\leq H=\BH(K)$ a finitely
generated Zariski dense subgroup. Then there is a constant
$m_1=m_1(\gC)$ such that for any symmetric generating set $\gS\ni
1$ of $\gC$, $\gS^{m_1}$ contains two independent elements which
generate a Zariski dense subgroup of $\BH$.
\end{thm}
\begin{rem}\label{rem:Z-dense}
The proof of Theorem \ref{thm:Z-dense} also shows that Theorem \ref{thm2} remains true, in characteristic zero, with the stronger conclusion that the independent
elements generate a Zariski dense subgroup of $\BG$.
\end{rem}
In order to obtain Theorem \ref{thm:Z-dense} one needs to slightly
modify the argument of Section \ref{sec:ping-pong} in a few places.
We will now indicate these modifications. For the sake of
simplicity, let us assume that $\BH$ is simple.
It is well known that $\BH$ admits two conjugate elements which
generate a Zariski dense subgroup. Indeed, one can take a regular
unipotent in $\BH$ and a conjugate lying in an opposite
parabolic (these unipotent elements can be taken in $\BH(\tilde{K})$,
where $\tilde{K}$ is a finite extension of $K$ over which $\BH$ is
isotropic). Let $\mathcal{A}$ be the subalgebra spanned by
$\text{Ad}(\BH)$ in $\text{End}(\mathfrak{h})$ where
$\mathfrak{h}$ denotes the Lie algebra of $\BH$, set
$$
F=\{ (g,h)\in\BH\times\BH:\text{Ad}(g)~\text{and}~\text{Ad}(h)~\text{do not generate the algebra}~\mathcal{A}\},
$$
and $E=\{ (g,h)\in\BH\times\BH:(g,hgh^{-1})\in F\}$.
It follows the algebraic variety $E$
is proper in $\BH\times\BH$. Let $E_1=\{ g\in \BH: (g,h)\in
E~\forall h\in\BH\}$, and for $g\in\BH$ let $E_2(g)=\{
h\in\BH:(g,h)\in E\}$. Then $E_1$ is a proper subvariety of $\BH$
and one easily checks that $\chi (E_2(g))$ is bounded independently
of $g$.
Note that in Theorem \ref{thm:Z-dense}, we did not assume that the field is a global field. In order to take
care of this issue we will use the specialization map introduced in Lemma \ref{lem:spec}. Without loss of generality,
we may pass to a subgroup of finite index in $\gC$. We will need the following lemma.
\begin{lem}\label{graphdensity}
Let $f: \Gamma \mapsto \BG$ be the specialization map from Lemma \ref{lem:spec}. Then the subgroup $\gD=\{(\gc, f(\gc)) \in \BH \times \BG\ / \gc \in \gC\}$ is not contained in any algebraic subset of the form $(V \times \BG) \cup (\BH \times W)$, where $V$ and $W$ are proper closed subvarieties of $\BH$ and $\BG$ respectively.
\end{lem}
\begin{proof}
This is obvious since $\gC$ is Zariski dense in $\BH$ and $f(\gC)$
is Zariski dense in $\BG$.
\end{proof}
When pursueing the argument of Section 6, we need to specify conditions on the elements of the generating set. These conditions are set on elements of $f(\gC) \in \BG$. We will now introduce new algebraic conditions directly on the elements of $\gC \in \BH$. Combining Lemma \ref{Bezout} with Lemma \ref{graphdensity} we see that given a set of non-trivial algebraic conditions in $\BH$ and another such set in $\BG$, there is an integer $N$ such that, for every generating set $\gS$ of $\gC$, there is a point $\gc \in \gS^N$ such that $\gc\in \BH$ and $f(\gc) \in \BG$ do not satisfy those conditions. This will be used repeatedly below. When no confusion may arise we will often say for instance that some element $A \in \gC$ acts proximally when we really mean that $f(A) \in \BG$ acts proximally on the representation variety used in Section 6.
The first modification needed in the argument of Section
\ref{sec:ping-pong} is in Proposition \ref{prop:very-proximal}
when we construct the very proximal element $X$. Instead of using
the same element $B_1$ which was used in the construction of the
very contracting element, we should use an element $B_1'$ which
satisfies
\begin{itemize}
\item $f(B_1')$ is in $(2n-1)$--general position with respect to $\{\overrightarrow{u_i}\}_{i-1}^n$ (like $f(B_1)$), and
\item $(B_1')^k\notin E_1A_2^{-1},~\forall k\leq 2n-1$.
\end{itemize}
Note that the choice of $B_1'$ depends on $A_2$, however, since
$\chi (E_1A_2^{-1})$ is independent of $A_2$ we can find $B_1'$ in a
fixed power of our generating set $\gS$. Retrospectively we should
also take the constant $r_6$ big enough so that the very
contracting element $A_2$ constructed in Proposition
\ref{prop:very-proximal} will have sufficiently small attracting
and repelling neighborhoods (i.e. that $|\ga_1/\ga_2|^{-r_6}$ will
be small enough) so that the element $X=(B_1')^kA_2$ (where $k$ is
some integer $\leq 2n-1$) becomes very proximal. Additionally, we
have to take the constant $s$ in Proposition \ref{sigma-two}
sufficiently large to guarantee that $f(X)$ is still semisimple.
The second change one has to do is in Step (4) when choosing the
appropriate conjugation of $X$. By the choice of $B_1'$ we know
that $X\notin E_1$. We take $B_2'$ which satsfies:
\begin{itemize}
\item $f(B_2')$ is in $(2n-1)^2$--general position with respect to the eigenvectors of $f(X)$ (again chosen as in Lemma \ref{nice-vectors}), and
\item $(B_2')^k\notin E_2(X),~\forall k\leq (2n-1)^2$.
\end{itemize}
Then, as in the previous section, if $Y=(B_2')^{k'}X(B_2')^{-k'}$
for some appropriate $k'\leq (2n-1)^2$ then $X^t$ and $Y^{t}$ are
independent for any $t\geq r_{13}'$ for some constant $r_{13}'$.
Finally, since $F$ is an algebraic subvariety of $\BH\times\BH$ and $\chi(Fx)$ is bounded independently of $x \in \BH\times\BH$, we may apply
Lemma \ref{Bezout} to the set $\{(X,Y)\}$ and the variety $F\cdot (X^{-r_{13}'}, Y^{-r_{13}'})$. Since $\{(X,Y)\}$ is not in $F$, it follows that $\{(X,Y)\}$ generates a group not contained in $F\cdot (X^{-r_{13}'}, Y^{-r_{13}'})$. Hence Lemma \ref{Bezout} yields some $t$ with $r_{13}'\leq t\leq r_{13}'+N(\chi(F))$ such that $(X^t,Y^t)\notin F$. Now since $X$ has infinite order, the
Zariski connected group $(\overline{\langle
X^t,Y^t\rangle}^Z)^\circ$ has positive dimension, and since it is
normalized by $X^t$ and $Y^t$, while
$\text{span}\{\text{Ad}(X^t),\text{Ad}(Y^t)\}=\mathcal{A}$ it
follows that $(\overline{\langle X^t,Y^t\rangle}^Z)^\circ$ is
normal in $\BH$. Since $\BH$ is assumed to be simple we derive
that $\langle X^t,Y^t\rangle$ is Zariski dense.\qed
\begin{thm}\label{thm:strong-version}
Let $\BG$ be a semisimple algebraic group defined over a field $K$ of characteristic zero, $\gC$ a finitely generated Zariski dense subgroup of
$\BG(K)$, and $V\subset\BG\times\BG$ a proper algebraic
subvariety. Then there is a constant $m=m(\gC,V)$ such that for
any generating set $\gS\ni 1$ of $\gC$, $\gS^m$ contains a pair of
independent elements $x,y$ with $(x,y)\notin V$.
\end{thm}
\begin{proof} The subset $\gS^{m_1}$ contains a pair $\{A,B\}$ of independent elements for some constant $m_1=m_1(\gC)$ given by Theorem \ref{thm:Z-dense}. This pair generates a Zariski dense subgroup of $\gC$. Hence $(1,A)$, $(1,B)$, $(A,1)$ and $(B,1)$ together generate a Zariski dense subgroup of $\BG \times \BG$. The set $V'=V \cup \{(x,y) | [x,y]=1\}$ is a proper closed algebraic subset of $\BG \times \BG$. By Lemma \ref{Bezout}, there exists another constant $m_2=m_2(V)$ such that
some word of length at most $m_2$ in those four generators lies
outside $V'$. This word has the form $(W_1(A,B),W_2(A,B))$ where
$W_1,W_2$ are bounded words in $A$ and $B$ that do not commute as
words in the free group. It follows that they generate a free
subgroup, hence form a pair of independent elements in
$\gS^{m_1m_2}$.
\end{proof}
\section{Some applications}\label{sec:applications}
In this section we draw some consequences of our main result.
\subsection{Uniform non-amenability and a uniform Cheeger
constant.}\label{subsec:u-n-a}
Recall that a group is called amenable if the regular
representation admits almost invariant vectors. It follows that if
a non-amenable group $\gC$ is generated by a finite set $\gS$ then
there is a positive constant $\gep(\gS )$ such that for any $f\in
L^2(\gC )$ there is some $\gs\in\gS$ for which $\|\rho (\gs
)(f)-f\|\geq\gep (\gS )\| f\|$, where $\rho$ denotes the left
regular representation, i.e. $\rho(\gc )(f)(x):=f(\gc^{-1}x)$.
Such an $\gep (\gS )$ is called a {\it Kazhdan constant for
$(\gS,\rho )$}. A finitely generated group $\gC$ is said to be
{\it uniformly non-amenable} if there is a positive Kazhdan
constant $\gep =\gep(\gC )>0$ for the regular representation which
is independent of the generating set $\gS$, i.e. if there is
$\gep>0$ such that for any generating set $\gS$ of $\gC$ and any
$f\in L^2(\gC )$ there is $\gs\in\gS$ for which $\|\rho (\gs
)(f)-f\|\geq \gep\| f\|$. It was observed by Y. Shalom \cite{Sha}
that Theorem \ref{thm1} implies:
\begin{thm}\label{thm:u-n-a}
A finitely generated non-amenable linear group is uniformly
non-amenable.
\end{thm}
\begin{proof}
The proof is an elaboration of the original proof by Von-Neumann
that a group which contains a non-abelian free subgroup is
non-amenable.
Let $\gC$ be a non-amenable linear group, and let $m=m(\gC )$ be
the constant from Theorem \ref{thm1}. Let $\gS$ be a generating set
of $\gC$ and let $x,y\in (\gS\cup\gS^{-1}\cup 1)^m$ be two
independent elements. Denote by $F_2=\langle x,y\rangle$ the
corresponding free subgroup. Choose a complete set $\{ c_i\}$ of
right coset representatives for $F_2$ in $\gC$, and write
$$
L^2(\gC )=\bigoplus L^2(F_2c_i).
$$
Let $f\in L^2(\gC )$, and let $f_i$ denote the restriction of $f$
to $F_2c_i$. Let $\gt_0$ be the Kazhdan constant for
$(\rho_{F_2},\{x,y\})$ then for any $i$ either $x$ or $y$ moves
$f_i$ by at least $\gt_0\| f_i\|$. Let
$$
f_x=\sum_{\|\rho (x)f_i-f_i\|\geq\gt_0\| f_i\|}f_i, ~~\text{and}~~
f_y=\sum_{\|\rho (y)f_i-f_i\|\geq\gt_0\| f_i\|}f_i
$$
Then either $\| f_x\|\geq\|f\|/\sqrt{2}$ or $\|
f_y\|\geq\|f\|/\sqrt{2}$. Without loss of generality let us assume
that $\| f_x\|\geq{\| f\|}/{\sqrt 2}$. It follows that
$$
\|\rho (x)f-f\|\geq\|\rho (x)f_x-f_x\|\geq\gt_0\|
f_x\|\geq\frac{\gt_0}{\sqrt{2}}\|f\|.
$$
Now write $x=\gs_1^{\gep_1}\cdot\ldots\cdot\gs_m^{\gep_m}$ where
$\gs_i\in\gS\cup\{ 1\}$ and $\gep_i=\pm 1$. Then by the triangle
inequality, if we let $\gs_0=1$ and $\gep_0=1$
$$
\|\rho (x)f-f\|\leq\sum_{i=1}^{m}\|\rho
(\gs_0^{\gep_0}\cdot\ldots\cdot\gs_i^{\gep_i})f-\rho
(\gs_0^{\gep_0}\cdot\ldots\cdot\gs_{i-1}^{\gep_{i-1}})f\|=
\sum_{i=1}^m\|\rho (\gs_i)f-f\|,
$$
and hence, for some $i$ we have $\|\rho(\gs_i
)f-f\|\geq\frac{\gt_0}{m\sqrt{2}}\| f\|$.
\end{proof}
Note that usually such groups do not admit a uniform Kazhdan
constant for arbitrary unitary representation, even if they have
property $(T)$ (see \cite{GZ}).
By considering $f$ to be a characteristic function of a finite
subset of $\gC$ and applying Theorem \ref{thm:u-n-a} we obtain the
following useful result. We denote by $|A|$ the number of elements
in $A$ and by $\bigtriangleup$ the operator of symmetric
difference between sets.
\begin{thm}\label{cor:boundary}
Let $\gC$ be a finitely generated non-virtually solvable linear
group. Then there is a positive constant $b=b(\gC )$ such that for
any generating set $\gS$ (not necessarily finite or symmetric) of
$\gC$, and any finite subset $A\subset\gC$ there is $\gs\in\gS$
such that:
$$
\frac{|\gs\cdot A\bigtriangleup A|}{|A|}\geq b.
$$
\end{thm}
Consider a graph $X$ and a finite subset $A\subset X$. The {\it
boundary} of $A$ is the set $\partial A$ of all vertices in $A$
which have at least one neighbor outside $A$. The {\it Cheeger
constant} $\mathcal{C}(X)$ is defined by
$$
\mathcal{C}(X)=\inf\frac{|\partial A|}{|A|},
$$
where $A$ runs over all finite subsets of $\text{vert}(X)$ when
$X$ is infinite, and over all subsets of size at most
$|\text{vert}(X)|/2$ when $X$ is finite. For a group $\gC$ and a
finite generating set $\gS$ we denote by $\mathcal{C}(\gC,\gS)$
the Cheeger constant of the Cayley graph of $\gC$ with respect to
$\gS$, and by $\mathcal{C}(\gC )$ the {\it uniform Cheeger
constant} of $\gC$:
$$
\mathcal{C}(\gC ):=\inf\{\mathcal{C}(\gC,\gS):\gS\text{~is a finite generating
set~}\}.
$$
In some places (c.f. \cite{Arz},\cite{Sha}) a group is called
uniformly non-amenable if it has a positive uniform Cheeger
constant. Clearly our definition of uniform non-amenability
implies this one\footnote{It is still not known whether these two
definitions are equivalent for a general finitely generated group.}:
\begin{cor}\label{cor:expanders}
A finitely generated non-virtually solvable linear group has a
positive uniform Cheeger constant.
\end{cor}
When $\mathcal{C}(X)>\gep$ the graph $X$ is said to be {\it
$\gep$--expander}. Hence Corollary \ref{cor:expanders} can be
reformulated as follows:
\begin{cor}
The family of all Cayley graphs corresponding to finite generating
sets of a given non-virtually solvable linear group $\gC$ form a
family of $\gd$--expanders for some constant $\gd=\gd (\gC )>0$.
\end{cor}
\subsection{Growth}\label{sub-sec:growth}
In \cite{EMO}, Eskin Mozes and Oh proved that any finitely
generated non-virtually solvable linear group has a uniform
exponential growth by showing that some bounded words in the
generators generate a free semigroup. As a consequence of Theorem
\ref{thm1} (more precisely of Theorem \ref{cor:boundary}) we
obtain:
\begin{thm}\label{thm:growth}
Let $\gC$ be a finitely generated non-virtually solvable linear
group. Then there is a constant $\lambda =\gl (\gC )>1$ such that
if $\gS$ is
any finite generating set of $\gC$, then $|\gS^n|\geq |\gS |\gl^{n-1},~\forall n\in\BN$.
\end{thm}
Since the proof is straightforward, we will omit it. One can
actually take $\gl =1+\frac{b}{2}$ where $b$ is the constant from
Theorem \ref{cor:boundary}.
\begin{rem}
Theorem \ref{thm:growth} improves Eskin-Mozes-Oh theorem in
several aspects:
\begin{itemize}
\item Unlike the situation in \cite{EMO}, the generating set $\gS$
in Theorem \ref{thm:growth} is not assumed to be symmetric, so
\ref{thm:growth} gives the uniform exponential growth for
semigroups rather than just for groups.
\item We didn't make the assumption from \cite{EMO} that the
characteristic of the field is $0$.
\item The estimate on the growth that we obtain is sharper; In
particular if the generating set is bigger the growth is faster.
This sharper estimate is important for applications,
\end{itemize}
\end{rem}
As another consequence we obtain that also the spheres have
uniform exponential growth, and moreover the size of each sphere
is at least $\frac{b}{2}$ times the size of the corresponding
ball. If $\gS\ni 1$ is a generating set for $\gC$ the sphere
$S(n,\gS )$ corresponding to $\gS$ is the set of all elements in
$\gC$ of distance exactly $n$ from $1$ in the Cayley graph, i.e.
$S(n,\gS )=\gS^n\setminus\gS^{n-1}$.
\begin{cor}
Let $\gC,\gS$ and $\gl=1+\frac{b}{2}$ be as in Theorem
\ref{thm:growth}. Then
$$
|S(n,\gS)|\geq\frac{b}{2}|\gS^{n-1}|\geq
\frac{b}{2}|\gS |(1+\frac{b}{2})^{n-2}.
$$
\end{cor}
One can derive many other variants of these results from Theorem
\ref{cor:boundary}. Here is another example:
\begin{exercise}
Let $\gC,\gS$ and $\gl$ be as above. There is a sequence
$\{\gs_i\}_{i \in \BN}$ of elements of $\gS$ such that for any
$n\in\BN$
$$
|\{\prod_{1\leq i_1<\ldots<i_k\leq
n}\gs_{i_1}\cdot\ldots\cdot\gs_{i_k}\}|\geq\gl^{n-1}.
$$
\end{exercise}
\subsection{Dense free subgroups, amenable actions and growth of
leaves.}\label{seb-sec:dense}
Theorem \ref{thm1} implies the following result from \cite{BG}
which answered a question of Carri\`{e}re and Ghys \cite{Ghys}:
\begin{thm}\label{thm:dense}
Let $G$ be a connected semisimple Lie group and $\gC\leq G$ a
dense subgroup. Then $\gC$ contains a dense free subgroup of rank
$2$.
\end{thm}
\begin{proof}
Let us assume for simplicity that $G$ is simple. The adjoint
representation $\text{Ad}:G\to\GL(\mathfrak{g})$ is irreducible
and by Burnside's theorem its image spans
$\text{End}(\mathfrak{g})$. It is well known that
$\text{End}(\mathfrak{g})$ is generated by two elements and that
these elements can be chosen in $\text{Ad}(G)$. Since
$\text{End}(\mathfrak{g})$ is finite dimensional it follows that
the set
$$
V=\{ (g_1,g_2)\in G\times G:
\text{Ad}(g_1)\text{~and~}\text{Ad}(g_2)\text{~generate~}\text{End}(\mathfrak{g})\}
$$
is Zariski open $G\times G$. By Theorem \ref{thm:strong-version}
and the remark following it there is a constant $m=m(\gC,V)$ such that if $\gS\ni 1$ is any
generating set for $\gC$ then $\gS^m$ contains independent
elements $x,y$ with $(x,y)\notin V$. Let $U\subset G$ be a
Zassenhaus neighborhood (c.f. \cite{raghunathan} 8.16), and let
$\gO$ be an identity neighborhood with $\gO^m\subset U$. Take
$\gS=\gC\cap\gO$. Since $G$ is connected and $\gC$ is dense, $\gS$
generates $\gC$, and therefore $\gS^m$ contains $x,y$ independent
with $(x,y)\notin V$ according to Theorem
\ref{thm:strong-version}. Now the connected component of the
identity in $\overline{\langle x,y\rangle}$ is normalized by $x,y$
and as $(x,y)\notin V$ it is normal in $G$, and by simplicity of
$G$ it is either $1$ or $G$. In other words $\langle x,y\rangle$
is either discrete or dense. However $\langle x,y\rangle$ is free
and hence non-nilpotent and since $x,y\in U$ it follows from
Zassenhaus' theorem that $\langle x,y\rangle$ is not discrete.
\end{proof}
Recall that one of the main motivation to prove Theorem
\ref{thm:dense} was the Connes--Sullivan conjecture which was
first proved by Zimmer:
\begin{cor}[Zimmer \cite{Zim}]
Let $\gC$ be a countable subgroup of a real Lie group $G$. Then
the action of $\gC$ on $G$ by left translations is amenable iff
the connected component of the identity of the closure of $\gC$ is
solvable.
\end{cor}
This corollary follows from Theorem \ref{thm:dense} by the
observation of Carri\`{e}re and Ghys that a non-discrete free
subgroup of $G$ cannot act amenably (see \cite{BG},\cite{BG1} for
more details and stronger results).
Another motivation was the result about the
polynomial--exponential dichotomy for the growth of leaves in
Riemannian foliations which was conjectured by Carri\`{e}re:
\begin{thm}[\cite{BG1}]\label{thm:foliations}
Let $\mathcal{F}$ be a Riemannian foliation on a compact manifold.
Then either the growth of any leaf in $\mathcal{F}$ is polynomial,
or the growth of a generic leaf is exponential.
\end{thm}
Theorem \ref{thm:foliations} can be considered as a foliated
version of the well known conjecture according to which the growth
of the universal cover of any compact Riemannian manifold is
either polynomial or exponential. The proof of
\ref{thm:foliations} relies on the following strengthening of
Theorem \ref{thm:dense} as well as some special argument for
solvable groups (see \cite{BG1} for more details):
\begin{thm}(\cite{BG})\label{thm:dense1}
Let $G$ be a connected semisimple Lie group, $\gC\leq G$ a dense
subgroup and $\gO_1,\ldots,\gO_n$ some $n$ open sets in $G$. Then
one can pick $x_i\in\gC\cap\gO_i,~i=1,\ldots,n$ which are
independent, i.e. generate a free group of rank $n$.
\end{thm}
In \cite{BG} Theorem \ref{thm:dense1} was the main result and
Theorem \ref{thm:dense} followed as a consequence. Let us show
that conversely it is possible to derive Theorem \ref{thm:dense1}
from Theorem \ref{thm:dense}. This way, Theorem \ref{thm:dense1}
will appear as a mere consequence of the main theorem of the
present paper, namely Theorem \ref{thm1}.
\textit{Proof that Theorem \ref{thm:dense} implies Theorem
\ref{thm:dense1}}. Let $F_2\leq\gC$ be a free subgroup of rank $2$
which is dense in $G$, and let $F_n$ be a subgroup of index $n-1$
in $F_2$. Then $F_n$ is a free group of rank $n$ which is still
dense in $G$. We will pick the $x_i$ in $F_n$ inductively as
follows. Suppose we picked already $x_1,\ldots,x_{i-1}$. Since
$F_n$ is dense and $\gO_i$ is open, $F_n$ is generated by
$F_n\cap\gO_i$. It follows that we can pick $x_i\in F_n\cap\gO_i$
such that the abelianization of $\langle x_1,\ldots,x_i\rangle$
has rank $i$; Indeed, look at the tensor of the abelanization of
$F_n$ with $\BQ$ and pick $x_i$ in the generating set
$F_n\cap\gO_i$ which is not in the $(i-1)$--dimensional
$\BQ$--subspace spanned by the images of $x_1,\ldots,x_{i-1}$. It
follows that $\langle x_1,\ldots,x_i\rangle$ is a free group whose
minimal number of generators is exactly $i$. Since a free group is
Hopfian it follows that $x_1,\ldots,x_i$ are independent. \qed
\medskip
We refer the reader to \cite{BG1} for an extension of
Theorems \ref{thm:dense} and \ref{thm:dense1} to a more general
setup.
| 28,543
|
\begin{document}
\maketitle
\begin{abstract}
In their response to the COVID-19 outbreak, governments face the dilemma to balance public health and economy. Mobility plays a central role in this dilemma because the movement of people enables both economic activity and virus spread. We use mobility data in the form of counts of travellers between regions, to extend the often-used SEIR models to include mobility between regions.
We quantify the trade-off between mobility and infection spread in terms of a single parameter, to be chosen by policy makers, and propose strategies for restricting mobility so that the restrictions are minimal while the infection spread is effectively limited. We consider restrictions where the country is divided into regions, and study scenarios where mobility is allowed within these regions, and disallowed between them. We propose heuristic methods to approximate optimal choices for these regions.
We evaluate the obtained restrictions based on our trade-off. The results show that our methods are especially effective when the infections are highly concentrated, e.g., around a few municipalities, as resulting from superspreading events that play an important role in the spread of COVID-19.
We demonstrate our method in the example of the Netherlands. The results apply more broadly when mobility data is available.
\end{abstract}
\section{Introduction}
The pandemic of COVID-19, caused by the coronavirus SARS-CoV-2, has, by mid-October 2020, infected more than 40 million people in over 200 countries and led to more than a million deaths. It is unlikely that the spread can be fully controlled in the near future and without the deployment of effective vaccines. Strategies are aimed at curbing exponential growth in case numbers and hospitalisations, predominantly to keep national health systems from becoming overburdened and to reduce infection pressure for people with a high risk of severe outcomes. Such strategies are limited to personal protection and hygiene, social distancing measures, reducing contacts and mixing/mobility.
Although the virus is present globally, all countries implement their own strategies and sets of measures.
At any given moment in the outbreak, there is a mix of countries and regions where the virus is temporarily under control, countries where the epidemic is decreasing and countries where the epidemic is increasing.
After an initial peak in cases, countries remain at risk for second and subsequent peaks, even when no cases are reported in the country for long periods of time. As in principle everybody is susceptible to some degree, not reaching herd immunity after the initial wave of infection leaves a large susceptible population that can sustain subsequent outbreaks \cite{anderson2020will}. These new outbreaks can be triggered by infected individuals entering the country from outside, as a result of increased global mobility. Nationally, sustained transmission at relatively low levels can lead to new large (exponentially growing) outbreaks after the initial peak because control measures are relaxed or behaviour changes with respect to (social, temporal and spatial) mixing and personal protection/social distancing. Mixing increases the number of new contact opportunities that an infected individual has in the population and reduced effectiveness of personal protection and social distancing increases the probability per contact of transmission. Combined, these effects can lead to more transmission. Increased mixing not only reflects larger groups of individuals, but also reflects contacts with individuals from a larger geographic range, allowing infected individuals to have contacts with people from regions where infection pressure may hitherto be (very) low, causing clusters of cases in new areas.
Mobility between areas plays a potentially important role in increasing transmission, but measures aimed at restricting mobility also have a potentially large controlling influence. Where, in the initial wave of infection, countries to a large extent imposed national mobility restrictions, the containment strategies for preventing subsequent waves of infection can perhaps be achieved by more regional or local mobility restrictions. This has the advantage of reducing the social and economic burden on society, but also has the risk that the restrictions may not be sufficiently effective and need to be scaled-up after all to a national level at some later point in time. It is, however, unclear how one could gauge the effectiveness of regional restrictions based on realistic mobility patterns specific to the country, balancing trade-offs between mobility and transmission. It is also unclear how large a `region' should be for effective containment and how different choices for recognizable regions (for example, administrative regions such as provinces, large cities, or postal code regions).
In this paper, we provide a framework to evaluate the effectiveness of regional strategies aimed at restricting mobility, allowing for a range of choices of how regions are characterised, using the Netherlands as a case study. This is essential to be able to determine the scale at which interventions can be effectively imposed or lifted and addresses one of a range of key modelling questions for COVID-19 and future pandemic outbreaks \cite{thompson2020key}. We base the framework on actual mobility patterns in the Netherlands. We distinguish between extreme situations where infection is distributed evenly between areas and situations where infection is highly concentrated in a restricted area, for example as a result of a superspreading event. We show that regions defined on the basis of mobility patterns provide better strategies than regions based on administrative characteristics, and that focusing on administrative regions therefore leads to sub-optimal strategies. We also quantify and explore the non-linear relation between mobility and outbreak size for a range of choices of trade-off between mobility and transmission.
\input{methodology.tex}
\input{results.tex}
\input{conclusion.tex}
\paragraph{\bf Acknowledgments.} We thank Paul van der Schoot for stimulating discussions. The work of MG and RvdH is supported by the Netherlands Organisation for Scientific Research (NWO) through the Gravitation {\sc Networks} grant 024.002.003. The work of HH, RvdH and NL is further supported by NWO through ZonMw grants 10430022010001 and 10430 03201 0011.
\printbibliography
\appendix
\section{Supplementary material}
\input{supplementary.tex}
\end{document}
| 129,146
|
\begin{document}
\bibliographystyle{amsplain}
\title{{Convexity of the Generalized Integral Transform and Duality Techniques }}
\author{
Satwanti Devi
}
\address{
Department of Mathematics \\
Indian Institute of Technology, Roorkee-247 667,
Uttarakhand, India
}
\email{ssatwanti@gmail.com}
\author{
A. Swaminathan
}
\address{
Department of Mathematics \\
Indian Institute of Technology, Roorkee-247 667,
Uttarakhand, India
}
\email{swamifma@iitr.ernet.in, mathswami@gmail.com}
\bigskip
\begin{abstract}
Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of
normalized analytic functions $f$ defined in the domain $|z|<1$
satisfying
\begin{align*}
{\rm Re\,} e^{i\phi}\left(\dfrac{}{}(1\!-\!\alpha\!+\!2\gamma)\!\left({f}/{z}\right)^\delta
+\left(\alpha\!-\!3\gamma+\gamma\left[\dfrac{}{}\left(1-{1}/{\delta}\right)\left({zf'}/{f}\right)+
{1}/{\delta}\left(1+{zf''}/{f'}\right)\right]\right)\right.\\
\left.\dfrac{}{}\left({f}/{z}\right)^\delta \!\left({zf'}/{f}\right)-\beta\right)>0,
\end{align*}
with the conditions $\alpha\geq 0$, $\beta<1$, $\gamma\geq 0$,
$\delta>0$ and $\phi\in\mathbb{R}$. Moreover, for
$0<\delta\leq\frac{1}{(1-\zeta)}$, $0\leq\zeta<1$, the class
$\mathcal{C}_\delta(\zeta)$ be the subclass of normalized
analytic functions such that
\begin{align*}
{\rm Re}{\,}\left(1/\delta\left(1+zf''/f'\right)+(1-1/\delta)\left({zf'}/{f}\right)\right)>\zeta,\quad |z|<1.
\end{align*}
In the present work, the sufficient conditions on $\lambda(t)$
are investigated, so that the generalized integral transform
\begin{align*}
V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t) \left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta},\quad |z|<1,
\end{align*}
carries the functions from
$\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ into
$\mathcal{C}_\delta(\zeta)$. Several interesting applications
are provided for special choices of $\lambda(t)$.
\end{abstract}
\subjclass[2000]{30C45, 30C55, 30C80}
\subjclass[2000]{30C45, 30C55, 30C80}
\keywords{
Duality techniques, Integral transforms, Convex functions,
Starlike functions, Hypergeometric functions, Hadamard Product
}
\maketitle
\pagestyle{myheadings} \markboth{Satwanti Devi and A. Swaminathan }{ Convexity of Generalized
Integral Transform and Duality Technique}
\section{introduction}
Let $\mathcal{A}$ be the class of all normalized analytic functions $f$ defined in the region $\mathbb{D}=\{
z\in\mathbb{C}: |z|<1\}$ with the condition $f(0)=f'(0)-1=0$ and $\mathcal{S}\subset\mathcal{A}$ be the class of all
univalent functions in $\mathbb{D}$.
We are interested in the following problem.
\begin{problem}\label{Prob1:Genl_operator}
Given $\lambda(t):[0,1]\!\rightarrow\! \mathbb{R}$ be a non-negative
integrable function with the condition $\int_0^1\lambda(t)
dt=1$, then for $f$ in a particular class of analytic functions, the generalized integral
transform defined by
\begin{align}\label{eq-weighted-integralOperator}
V_{\lambda}^\delta(f)(z):=\left(\int_0^1 \lambda(t) \left(\dfrac{f(tz)}{t}\right)^\delta dt\right)^{1/\delta},
\quad \delta>0\quad {\rm and}\quad z\in\mathbb{D}
\end{align}
is in one of the subclasses of ${\mathcal{S}}$.
\end{problem}
This problem, for the case $\delta=1$ was first stated by R. Fournier and S. Ruscheweyh \cite{FourRusExtremal} by examining the
characterization of two extremal problems. They
considered the functions $f$ in the class ${\mathcal{P}}_{\beta}$, where
\begin{align*}
{\mathcal{P}}_{\beta} = \left\{ \dfrac{}{} f\in {\mathcal{A}}: {\rm Re\,} \left(\dfrac{}{}e^{i\alpha}(f'(z)-\beta)\right)>0, \quad
\alpha\in {\mathbb{R}}, \quad z\in{\mathbb{D}}\right\}.
\end{align*}
such that the integral operator $V_{\lambda}(f)(z): V_{\lambda}^1(f)(z)$ is in the class ${\mathcal{S}^{\ast}}$ of functions that map
${\mathbb{D}}$ onto domain that are starlike with respect to origin using duality techniques. Same problem was solved by
R.M. Ali and V.Singh \cite{Ali} for functions $f$ in the class ${\mathcal{P}}_{\beta}$ so that the
integral operator $V_{\lambda}(f)(z)$ is in the class ${\mathcal{C}}$ of functions that map ${\mathbb{D}}$ onto domain that are convex.
The integral operator $V_{\lambda}(f)(z)$ contains some of the
well-known operator such as Bernardi, Komatu and Hohlov as its
special cases for particular choices of $\lambda(t)$, which has
been extensively studied by various authors (for details see
\cite{Ali, saigo, DeviPascu} and references therein).
Generalization of the class ${\mathcal{P}}_{\beta}$ for studying the above problem with reference to the operator $V_{\lambda}(f)(z)$ were
considered by several researchers in the recent past and interesting applications were obtained. For
most general result in this direction, see \cite{DeviPascuOrder} and references therein.
The integral operator \eqref{eq-weighted-integralOperator} and
its generalization was considered in the work of I. E.
Bazilevi\v c \cite{BazilevicIntegOper} (for more details see
\cite{Aghalary, SinghIntOperator}). Problem $\ref{Prob1:Genl_operator}$ for the generalized integral operator
$V_{\lambda}^\delta(f)(z)$ relating starlikeness was investigated by A. Ebadian et al. in \cite{Aghalary} by considering
the class
{\small{
\begin{align*}
P_\alpha(\delta,\beta):=
\left\{f\in\mathcal{A}\,,\,\exists\,\phi\in\mathbb{R}:\,{\rm Re\,} e^{i\phi}
\left((1-\alpha)\left(\dfrac{f}{z}\right)^\delta
+\alpha\left(\dfrac{f}{z}\right)^\delta \left(\dfrac{zf'}{f}\right)-\beta\right)>0,\,z\in\mathbb{D}\right\}
\end{align*}}}
with $\alpha \geq0$, $\beta<1$ and $\delta>0$.
The authors of the present work have generalized the starlikeness criteria \cite{DeviGenlStar}
by considering the following subclass of ${\mathcal{S}}^{\ast}$
\begin{align}\label{eq-gener:starlike-related:classes}
f\in \mathcal{S}^\ast_s(\zeta)\,\,\Longleftrightarrow\,\,
z^{1-\delta}f^{\delta}\in\mathcal{S}^\ast(\xi),
\end{align}
for $\xi=1-\delta+\delta\zeta$ and $0\leq\xi<1$ where
$\mathcal{S}^{\ast}(\xi)$ is the class having the analytic characterization
\begin{align*}
{\rm{Re}}{\,} \left(\dfrac{zf^\prime}{f}\right)>\xi,\quad 0\leq\xi<1, \quad z\in\mathbb{D}.
\end{align*}
Note that $\mathcal{S}^{\ast}: =\mathcal{S}^{\ast}(0)$.
In \cite{DeviGenlStar}, this problem was investigated
by considering the integral operator acting on the most generalized class of ${\mathcal{P}}_{\beta}$, related to the present context,
which is defined as follows.
{\small{
\begin{align*}
\mathcal{W}_{\beta}^\delta(\alpha,\gamma)\!:\!=\!\left\{\!f\!\in\!\mathcal{A}:
{\rm Re\,} e^{i\phi}\left((1\!-\!\alpha\!+\!2\gamma)\!\left(\frac{f}{z}\right)^\delta
\!+\!\left(\alpha\!-\!3\gamma\!+\!\gamma\left[\left(1\!-\!\frac{1}{\delta}\right)\left(\frac{zf'}{f}\right)\!+\!
\frac{1}{\delta}\left(1\!+\!\frac{zf''}{f'}\right)\right]\right)\right.\right.\\
\left.\left.\left(\frac{f}{z}\right)^\delta \!\left(\frac{zf'}{f}\right)-\beta\right)>0,
\,\,z\in\mathbb{D},\,\,\phi\in\mathbb{R}\right\}.
\end{align*}}}
\noindent Here, $\alpha \geq 0$, $\beta<1$, $\gamma\geq 0$ and $\phi\in {\mathbb{R}}$. Note that
$\mathcal{W}_{\beta}^\delta(\alpha,0)\equiv P_\alpha(\delta,\beta)$ is the class considered by A. Ebadian et al in \cite{Aghalary},
$R_\alpha(\delta,\beta):\equiv \mathcal{W}_{\beta}^\delta(\alpha+\delta+\delta\alpha,\delta\alpha)$ is a closely related class and
$\mathcal{W}_{\beta}^1(\alpha,\gamma)\equiv\mathcal{W}_{\beta}(\alpha,\gamma)$ introduced by R.M. Ali et al in \cite{AbeerS}.
As the investigation of this generalization provided fruitful results, we are interested in considering further geometric properties
of the generalized integral operator given by \eqref{eq-weighted-integralOperator} for $f\in \mathcal{W}_{\beta}^\delta(\alpha,\gamma)$.
Motivated, by the well-known Alexander theorem \cite{DU},
\begin{align*}
f\in\mathcal{C}(\xi)\Longleftrightarrow zf '\in\mathcal{S}^\ast(\xi),
\end{align*}
where ${\mathcal{C}}(0)={\mathcal{C}}$,
we consider the subclass
\begin{align}\label{iff:relation:convex+starlike}
f\in \mathcal{C}_\delta(\zeta)\Longleftrightarrow
(z^{2-\delta}f^{\delta-1}f')\in\mathcal{S}^\ast(\xi),
\end{align}
where $\xi:=1-\delta+\delta\zeta$ with the conditions
$1-\frac{1}{\delta}\leq\zeta<1$, $0\leq\xi<1$ and
$\delta\geq1$. In the sequel, the term $\xi$ is used to denote
$(1-\delta+\delta\zeta)$. From the above expression, it is easy
to observe that the class $\mathcal{C}_\delta(\zeta)$ and
$\mathcal{C}(\xi)$ are equal, when $\delta=1$.
The class $\mathcal{C}_\delta(\zeta)$ given in \eqref{iff:relation:convex+starlike}
is related to the class of $\alpha$- convex of order $\zeta$ $(0\leq\zeta<1)$ that were
introduced in the work of P. T. Mocanu \cite{MocanuAlphaConvex} and
defined analytically as
\begin{align*}
{\rm Re}{\,}\left(\left(1-\alpha\right)\left(\dfrac{zf'(z)}{f(z)}\right)+\alpha\left(1+\dfrac{zf''(z)}{f'(z)}\right)\right)>\zeta,
\quad (1-\zeta)\leq\alpha<\infty.
\end{align*}
Clearly the class $\mathcal{C}_\delta(\zeta)$ is nothing but the
subclass of $\mathcal{S}$ consisting of ${1}/{\delta}\,$-
convex functions of order $\zeta$.
Having provided all the required information from the literature, in what follows, we obtain sharp estimates
for the parameter $\beta$ so that the generalized
integral operator \eqref{eq-weighted-integralOperator}
maps the function from $\mathcal{W}_\beta^\delta(\alpha,\gamma)$ into
$\mathcal{C}_\delta(\zeta)$, where
$0<\delta\leq\frac{1}{(1-\zeta)}$ and $0\leq\zeta<1$.
Duality techniques, given in \cite{Rus} provide the platform for the entire study of this manuscript. One of the particular tool in this regard
is the convolution or Hadamard product of two
functions
\begin{align*}
f_1(z)=\sum_{n=0}^\infty a_n z^n\quad {\rm and}\quad f_2(z)=\sum_{n=0}^\infty b_n z^n,\quad z\in\mathbb{D},
\end{align*}
given by
\begin{align*}
(f_1\ast f_2)(z)\!=\!\displaystyle\sum_{n=0}^{\infty}a_nb_n
z^n.
\end{align*}
Furthermore, consider the complex parameters $c_i$ $(i=0,1,
\ldots,p)$ and $d_j$ $(j=0,1,\ldots,q)$ with
$d_j\neq0,-1,\ldots$ and $p\leq q+1$. Then, in the region
$\mathbb{D}$, the generalized hypergeometric function is given
by
\begin{align*}
{\,}_{p}F_q\left(\!\!\!\!\!
\begin{array}{cll}&\displaystyle c_1,\ldots,c_p
\\
&\displaystyle d_1,\ldots,d_q
\end{array};z\right)
=\sum_{n=0}^\infty
\dfrac{(c_1)_n\ldots,(c_p)_n}{(d_1)_n\ldots,(d_q)_nn!}z^n,\quad z\in\mathbb{D},
\end{align*}
that can also be represented as
$_{p}F_{q}(c_1,\ldots,c_p;d_1,\ldots,d_q;z)$ or $_{p}F_{q}$. In
particular, $_{2}F_{1}$ is the well-known Gaussian
hypergeometric function. For any natural number $n$, the
Pochhammer symbol or shifted factorial $(\varepsilon)_n$ is
defined as $(\varepsilon)_0=1$ and
$(\varepsilon)_n=\varepsilon(\varepsilon+1)_{n-1}$.
The paper is organized as follows: Necessary and sufficient conditions are
obtained in Section \ref{Sec-Gener:Convex-Main:Result} that
ensures
$V_\lambda^\delta(\mathcal{W}_\beta^\delta(\alpha,\gamma))\!\subset\!
\mathcal{C}_\delta(\zeta)$. The simpler sufficient criterion
are derived in Section \ref{Sec-Gener:Convex-Suff:Cond}, which
are further implemented to find many interesting applications involving various integral operators
for special choices of $\lambda(t)$.
\section{preliminaries}
The parameters $\mu,\,\nu\geq0$ introduced in \cite{AbeerS} are
used for further analysis that are defined by the relations
\begin{align}\label{eq-mu+nu}
\mu\nu=\gamma\quad \text{and}\quad \mu+\nu=\alpha-\gamma.
\end{align}
Clearly $\eqref{eq-mu+nu}$ leads to two cases.
\begin{itemize}
\item[{\rm{(i)}}] $\gamma=0 \, \Longrightarrow \mu=0,\,
\nu=\alpha \geq 0$.
\item[{\rm{(ii)}}] $\gamma>0 \, \Longrightarrow \mu>0,\,
\nu>0$.
\end{itemize}
Define the auxiliary function
\begin{align}\label{eq-Gener:convex-psi:munu}
\psi_{\mu,\nu}^\delta(z):=\sum_{n=0}^\infty\dfrac{\delta^2}{(\delta+n\mu)(\delta+n\nu)}z^n
=\int_0^1\int_0^1\dfrac{1}{(1-u^{\nu/\delta/}v^{\mu/\delta}z)}dudv,
\end{align}
which by a simple computation gives
\begin{align}\label{eq-Gener:convex-Phi:munu}
\Phi_{\mu,\nu}^\delta(z):=\left(z\psi_{\mu,\nu}^\delta(z)\right)'=\sum_{n=0}^\infty\dfrac{(n+1)\delta^2}{(\delta+n\nu)(\delta+n\mu)}z^n
\end{align}
\begin{align}\label{eq-Gener:convex-Upsilon:munu}
{\rm and}\quad\quad\Upsilon_{\mu,\nu}^\delta(z):=\left(z\left(z\psi_{\mu,\nu}^\delta(z)\right)'\right)'=
\sum_{n=0}^\infty\dfrac{(n+1)^2\delta^2}{(\delta+n\nu)(\delta+n\mu)}z^n.
\end{align}
Taking $\gamma=0$ $(\mu=0,\,\nu=\alpha\geq0)$, let
$q_{0,\alpha}^\delta(t)$ be the solution of the differential
equation
\begin{align}\label{eq-generalized:convex-q:gamma0}
\dfrac{d}{dt}\left(t^{\delta/\alpha}q_{0,\alpha}^\delta(t)\right)=\dfrac{\delta t^{\delta/\alpha-1}}{\alpha}
\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{(1-\xi\left(1+t\right))}{(1-\xi)\left(1+t\right)^2}
+\left(\dfrac{1}{\delta}\right)\dfrac{(1-t-\xi\left(1+t\right))}{(1-\xi)\left(1+t\right)^3}\right),
\end{align}
with the initial condition $q_\alpha^\delta(0)=1$. Then the
solution of \eqref{eq-generalized:convex-q:gamma0} is given by
\begin{align*}
q_{0,\alpha}^\delta(t)
=\dfrac{\delta t^{-\delta/\alpha}}{\alpha}
\int_0^t\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{(1-\xi\left(1+s\right))}{(1-\xi)\left(1+s\right)^2}
+\left(\dfrac{1}{\delta}\right)\dfrac{(1-s-\xi\left(1+s\right))}{(1-\xi)\left(1+s\right)^3}\right)s^{\delta/\alpha-1}ds.
\end{align*}
Also, for the case $\gamma>0$ $(\mu>0,\,\nu>0)$, let
$q_{\mu,\nu}^\delta(t)$ be the solution of the differential
equation
{\small{
\begin{align}\label{eq-Gener:Convex-q:gamma>0}
\dfrac{d}{dt}\left(t^{\delta/\nu}q_{\mu,\nu}^\delta(t)\right)=\dfrac{\delta^2 t^{\delta/\nu-1}}{\mu\nu}
\int_0^1\!\!\left(\!\left(1\!-\!\dfrac{1}{\delta}\right)\dfrac{(1\!-\!\xi(1\!+\!st))}{(1\!-\!\xi)(1\!+\!st)^2}\!+\!
\left(\dfrac{1}{\delta}\right)\dfrac{(1\!-\!st\!-\!\xi(1\!+\!st))}{(1\!-\!\xi)(1\!+\!st)^3}\right)s^{\delta/\mu-1}ds,
\end{align}}}
with the initial condition $q_{\mu,\nu}^\delta(0)=1$. Then the
solution of \eqref{eq-Gener:Convex-q:gamma>0} is given as
{\small{
\begin{align}\label{eq-Gener:Convex-Integral:q:gamma>0}
q_{\mu,\nu}^\delta(t)\!=\!\dfrac{\delta^2}{\mu\nu}\!\int_0^1\!\!\!\!\int_0^1\!\left(\!\left(1\!-\!\dfrac{1}{\delta}\right)
\dfrac{(1\!-\!\xi\left(1\!+\!trs\right))}{(1\!-\!\xi)\left(1\!+\!trs\right)^2}\!+\!
\left(\dfrac{1}{\delta}\right)\dfrac{(1\!-\!trs\!-\!\xi\left(1\!+\!trs\right))}
{(1\!-\!\xi)\left(1+trs\right)^3}\right)r^{\delta/\nu-1}s^{\delta/\mu-1}drds,
\end{align}}}
Furthermore, for given $\lambda(t)$ and $\delta>0$, we
introduce the functions
\begin{align}\label{eq-weighted:Lambda}
\Lambda_\nu^\delta(t):=\displaystyle\int_t^1\dfrac{\lambda(s)}{s^{{\delta}/{\nu}}}ds,\quad \nu>0,
\end{align}
and
\begin{align}\label{eq-weighted:Pi}
\Pi_{\mu,\nu}^\delta(t):=
\left\{\!
\begin{array}{cll}\displaystyle \int_t^1\!\!\dfrac{\Lambda_\nu^\delta(s)}{s^{{\delta}/{\mu}-{\delta}/{\nu}+1}}ds
&\quad\gamma\!>\!0 \,(\mu\!>\!0,\nu\!>\!0),\\\\
\Lambda_\alpha^\delta(t)
&\quad\gamma\!=\!0\,(\mu\!=\!0,\nu\!=\!\alpha\!\geq\!0)
\end{array}\right.
\end{align}
which are positive on $t\in(0,1)$ and integrable on
$t\in[0,1]$.
For $\delta=1$, these information coincide with the one given
in \cite{MahnazC}. \eqref{eq-weighted:Lambda} and \eqref{eq-weighted:Pi} are also considered in \cite{DeviGenlStar} and \cite{DeviGenlUniv}.
In \cite{DeviGenlStar}, the investigations are related to
$V_\lambda^\delta(f)(z)\in\mathcal{S}^\ast_s(\zeta)$, whenever $f\in\mathcal{W}_\beta^\delta(\alpha,\gamma)$, whereas
various other inclusion properties, in particular,
$V_\lambda^\delta(f)(z)\in\mathcal{W}_{\beta_1}^{\delta_1}(\alpha_1,\gamma_1)$, whenever $f\in\mathcal{W}_{\beta_2}^{\delta_2}(\alpha_2,\gamma_2)$
are investigated in \cite{DeviGenlUniv}.
\section{main results}\label{Sec-Gener:Convex-Main:Result}
The following result establishes both the necessary and
sufficient conditions that ensure
$F_\delta(z):=V_{\lambda}^\delta(f)(z)\in\mathcal{C}_\delta(\zeta)$,
whenever $f\in\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$.
\begin{theorem}\label{Thm-Gener:Convex-Mainresult}
Let $\mu\!\geq\!0$, $\nu\!\geq\!0$ are given by the relation in
\eqref{eq-mu+nu} and
$\left(1-\frac{1}{\delta}\right)\leq\zeta\leq
\left(1-\frac{1}{2\delta}\right)$ where $\delta\geq1$. Let
$\beta\!<\!1$ satisfy the condition
\begin{align}\label{Beta-Cond-Generalized:Convex}
\dfrac{\beta-\frac{1}{2}}{1-\beta}=-\int_0^1 \lambda(t) q_{\mu,\nu}^\delta(t) dt,
\end{align}
where $q_{\mu,\nu}^\delta(t)$ is defined by the differential
equation \eqref{eq-generalized:convex-q:gamma0} for $\gamma=0$
and \eqref{eq-Gener:Convex-q:gamma>0} for $\gamma>0$. Further
assume that the functions given in \eqref{eq-weighted:Lambda}
and \eqref{eq-weighted:Pi} attains
\begin{align*}
\lim_{t\rightarrow 0^+} t^{{\delta}/{\nu}}\Lambda_\nu^\delta(t)\rightarrow 0\quad{\rm and}\quad
\lim_{t\rightarrow 0^+}t^{{\delta}/{\mu}}\Pi_{\mu,\nu}^\delta(t)\rightarrow 0.
\end{align*}
Then for $f(z)\in\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$,
the function $F_\delta:=V_\lambda^\delta(f(z))\!\in\!
\mathcal{C}_\delta(\zeta)$ iff,
$\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)(z)\geq0$, where
\begin{align*}
\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)(z)\!:=\!\left\{\!\!
\begin{array}{cll}\displaystyle \int_0^1 t^{{\delta}/{\mu}-1}\Pi_{\mu,\nu}^\delta(t){\,} h_{\xi,\delta,z}(t)dt,\quad
&\gamma>0{\,}(\mu>0,{\,}\nu>0),\\\\
\displaystyle\int_0^1 t^{{\delta}/{\alpha}-1}\Lambda_\alpha^\delta(t){\,} h_{\xi,\delta,z}(t)dt,\quad
&\gamma=0{\,}(\mu=0,{\,}\nu=\alpha\geq0),
\end{array}\right.
\end{align*}
and
\begin{align*}
h_{\xi,\delta,z}(t):=\left(1\!-\!\dfrac{1}{\delta}\right)\!
\!\left({\rm Re}{\,}\dfrac{h_\xi(tz)}{tz}\!-\!\dfrac{1\!-\!\xi(1\!+\!t)}{(1\!-\!\xi)(1\!+\!t)^2}\right)\!+\!\left(\dfrac{1}{\delta}\right)\!
\left({\rm Re}{\,}h_\xi'(tz)\!-\!\dfrac{1\!-\!t\!-\!\xi(1\!+\!t)}{(1\!-\!\xi)(1\!+\!t)^3}\right)
\end{align*}
for the function
\begin{align}\label{eq-h(z)-extremal-starlike}
h_\xi(z):=z\left(\dfrac{1+\frac{\epsilon+2\xi-1}{2(1-\xi)}z}{(1-z)^2}\right),\quad |\epsilon|=1
\end{align}
and $\xi:=1-\delta(1-\zeta)$, $0\leq\xi\leq1/2$. The value of
$\beta$ is sharp.
\end{theorem}
\begin{proof}
From \eqref{iff:relation:convex+starlike}, it is clear that
\begin{align}\label{iff:relation:convex+starlike-integral}
F_\delta\in\mathcal{C}_\delta(\zeta)\Longleftrightarrow\left(z^{2-\delta}(F_\delta)^{\delta-1}F_\delta'\right)\in\mathcal{S}^\ast(\xi)
\end{align}
where $\xi:=1-\delta(1-\zeta)$. Thus, to prove
$\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)\geq 0$ using the
given hypothesis, it is required to show that the function
$z^{2-\delta}(F_\delta)^{\delta-1}F_\delta'$ is univalent and
satisfy the order of starlikeness condition, and conversely.
Let
{\small{
\begin{align}\label{eq-generalized:starlike-H}
H(z):=(1\!-\!\alpha\!+\!2\gamma)\!\left(\frac{f}{z}\right)^\delta
\!+\!\left(\alpha\!-\!3\gamma+\gamma\left[\left(1\!-\!\frac{1}{\delta}\right)\left(\frac{zf'}{f}\right)\!+\!
\frac{1}{\delta}\left(1\!+\!\frac{zf''}{f'}\right)\right]\right)
\left(\frac{f}{z}\right)^\delta\!\left(\frac{zf'}{f}\right).
\end{align}}}
Using the relation \eqref{eq-mu+nu} in
\eqref{eq-generalized:starlike-H} gives
\begin{align*}
H(z)
=\dfrac{\mu\nu}{\delta^2}z^{1-\delta/\mu}\left(z^{\delta/\mu-\delta/\nu+1} \left(
z^{\delta/\nu}\left(\dfrac{f}{z}\right)^\delta\right)'\right)'.
\end{align*}
Further, set $G(z)=(H(z)-\beta)/(1-\beta)$, then there exist
some $\phi\in\mathbb{R}$, such that ${\rm
Re}\left(e^{i\phi}{\,}G(z)\right)>0$. Hence by the duality
principle \cite[p. 22]{Rus}, we may confine to the function
$f(z)$ for which $G(z)={(1+xz)}/{(1+yz)}$, where $|x|=|y|=1$,
which directly implies
\begin{align*}
\dfrac{\mu\nu}{\delta^2}{\,} z^{1-\delta/\mu}\left(z^{\delta/\mu-\delta/\nu+1}
\left( z^{\delta/\nu}\left(\dfrac{f}{z}\right)^\delta\right)'\right)'
=(1-\beta)\dfrac{1+xz}{1+yz}+\beta,
\end{align*}
or equivalently,
\begin{align}\label{eq-generalized:starlike-f/z:series}
\left(\frac{f(z)}{z}\right)^\delta
&=\dfrac{\delta^2}{\mu\nu z^{{\delta}/{\nu}}}\left(\int_0^z\dfrac{1}{\eta^{{\delta}/{\mu}-{\delta}/{\nu}+1}}
\left(\int_0^\eta\dfrac{1}{\omega^{1-{\delta}/{\mu}}}\left((1-\beta)\dfrac{1+x\omega}{1+y\omega}+\beta\right)
d\omega\right)d\eta\right)\\
&=\beta+(1\!-\!\beta)\left(\left(\dfrac{1\!+\!xz}{1\!+\!yz}\right)\ast\sum_{n=0}^\infty\dfrac{\delta^2z^n}{(\delta+n\nu)(\delta+n\mu)}\right).
\end{align}
If $A(z)$ is taken as $\displaystyle\left(\dfrac{f(z)}{z}\right)^{\delta}$, then using
\eqref{eq-Gener:convex-psi:munu}, \eqref{eq-Gener:convex-Phi:munu} and \eqref{eq-Gener:convex-Upsilon:munu},
in $A(z)$, $zA'(z)$ and $z(zA'(z))'$ respectively, gives
\begin{align}\label{eq-Gener:Convex-zf/z-Double:Deriv:Convol}
\left(z\left(z\left(\frac{f(z)}{z}\right)^\delta\right)^\prime\right)^\prime
=\left(\beta+(1-\beta)
\left(\dfrac{1+xz}{1+yz}\right)\right)\ast\Upsilon_{\mu,\nu}^\delta(z).
\end{align}
Since
\begin{align}\label{eq-Gener:Convex-Simplified-zf/z:Deriv}
\left(z\left(\dfrac{f(z)}{z}\right)^\delta\right)'=
(1-\delta)\left(\dfrac{f(z)}{z}\right)^\delta+\delta\left(\dfrac{f(z)}{z}\right)^\delta\left(\dfrac{zf'(z)}{f(z)}\right),
\end{align}
this gives
\begin{align}\label{eq-Gener:Convex-Simpli-zf/z:Double:Deriv}
\left(z\left(\dfrac{f(z)}{z}\right)^\delta\left(\dfrac{zf'(z)}{f(z)}\right)\right)^\prime&=
\left(1-\dfrac{1}{\delta}\right)\left(z\left(\dfrac{f(z)}{z}\right)^\delta\right)'+
\dfrac{1}{\delta}\left(z\left(z\left(\dfrac{f(z)}{z}\right)^\delta\right)^\prime\right)^\prime,
\end{align}
taking the logarithmic derivative on both sides of the integral operator
\eqref{eq-weighted-integralOperator} and differentiating further with a simple computation involving
\eqref{eq-Gener:Convex-zf/z-Double:Deriv:Convol}
gives
\newline
$\displaystyle
\left(\!z\!\left(\!\dfrac{z(F_\delta)'}{F_\delta}\!\right)\!\!\left(\dfrac{F_\delta}{z}\right)^\delta\right)^\prime
$
\begin{align*}
\!=\!(1\!-\!\beta)\!\left(\!\int_0^1\!\!\!\lambda(t)\!\left(\!\left(\!1\!-\!\dfrac{1}{\delta}\!\right)\!\Phi_{\mu,\nu}^\delta(tz)
\!+\!\left(\!\dfrac{1}{\delta}\!\right)\!\Upsilon_{\mu,\nu}^\delta(tz)\!\right)\!dt\!+\!\dfrac{\beta}{(1\!-\!\beta)}\right)
\ast\left(\dfrac{1+xz}{1+yz}\right).
\end{align*}
The Noshiro-Warschawski's Theorem (for details see\cite[Theorem
2.16]{DU}) states that the function
$z^{2\!-\!\delta}\!(F_\delta)^{\delta\!-\!1}\!(F_\delta)'$
defined in the region $\mathbb{D}$ is univalent if
$\left(z^{2-\delta}(F_\delta)^{\delta-1}(F_\delta)'\right)'$ is
contained in the half plane not containing the origin. Hence,
from the result based on duality principle \cite[Pg. 23]{Rus},
it follows that
$0\neq\left(z\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\left(\dfrac{F_\delta}{z}\right)^\delta\right)^\prime$
\newline
is true if, and only if
\begin{align*}
{\rm Re}{\,}(1-\beta)\left(\int_0^1\lambda(t)\left(\left(1-\dfrac{1}{\delta}\right)\Phi_{\mu,\nu}^\delta(tz)
+\left(\dfrac{1}{\delta}\right)\Upsilon_{\mu,\nu}^\delta(tz)\right)dt+\dfrac{\beta}{(1-\beta)}\right)>\dfrac{1}{2}
\end{align*}
or equivalently,
\begin{align*}
{\rm Re}{\,}(1-\beta)\left(\int_0^1\lambda(t)\left(\left(1-\dfrac{1}{\delta}\right)\Phi_{\mu,\nu}^\delta(tz)
+\left(\dfrac{1}{\delta}\right)\Upsilon_{\mu,\nu}^\delta(tz)\right)dt+\dfrac{\beta-\frac{1}{2}}{(1-\beta)}\right)>0.
\end{align*}
Now, substituting \eqref{Beta-Cond-Generalized:Convex} in the
above inequality implies
\begin{align}\label{eq-Gener:Convex-Phi+Upsilon-q}
{\rm Re}{\,}\int_0^1\lambda(t)\left(\left(1-\dfrac{1}{\delta}\right)\Phi_{\mu,\nu}^\delta(tz)
+\left(\dfrac{1}{\delta}\right)\Upsilon_{\mu,\nu}^\delta(tz)- q_{\mu,\nu}^\delta(t)\right)dt>0.
\end{align}
From equation \eqref{eq-Gener:convex-Phi:munu} and
\eqref{eq-Gener:convex-Upsilon:munu}, it is easy to see that
\begin{align*}
\left(1-\dfrac{1}{\delta}\right)\Phi_{\mu,\nu}^\delta(tz)
+\left(\dfrac{1}{\delta}\right)\Upsilon_{\mu,\nu}^\delta(tz)
=\sum_{n=0}^\infty\dfrac{\delta (n+1)(n+\delta)(tz)^n}{(\delta+n\nu)(\delta+n\mu)},
\end{align*}
whose integral representation is given as
{\small{
\begin{align}\label{eq-generalized-Phi-Upsilon_munu-series}
\left(1\!-\!\dfrac{1}{\delta}\right)\Phi_{\mu,\nu}^\delta(tz)
\!+\!\left(\dfrac{1}{\delta}\right)\Upsilon_{\mu,\nu}^\delta(tz)=\dfrac{\delta^2}{\mu\nu}\!\int_0^1\!\!\!\int_0^1\!
\left(
\dfrac{\left(1\!-\!\frac{1}{\delta}\right)}{\left(1\!-\!trsz\right)^2}+\dfrac{\frac{1}{\delta}(1\!+\!trsz)}
{\left(1-trsz\right)^3}\right)r^{\delta/\nu-1}s^{\delta/\mu-1}drds.
\end{align}}}
Thus, using \eqref{eq-Gener:Convex-Integral:q:gamma>0} and
\eqref{eq-generalized-Phi-Upsilon_munu-series} in
\eqref{eq-Gener:Convex-Phi+Upsilon-q} and on further using the
fact that ${\rm Re\,} \left(\frac{1}{1-rstz}\right)^2\geq
\frac{1}{(1+rst)^2}$ for $z\in\mathbb{D}$, directly implies
{\small{
\begin{align*}
{\rm Re}{\,}\int_0^1&\lambda(t)\left(\!\int_0^1\!\!\!\int_0^1\!\left(\left(1\!-\!\dfrac{1}{\delta}\right)
\dfrac{1}{\left(1\!-\!trsz\right)^2}+\left(\dfrac{1}{\delta}\right)\dfrac{1\!+\!trsz}
{\left(1-trsz\right)^3}\right)r^{\delta/\nu-1}s^{\delta/\mu-1}drds\right.\\
&\left.-\!\int_0^1\!\!\!\int_0^1\!\left(\!\left(\!1\!-\!\dfrac{1}{\delta}\!\right)
\dfrac{1\!-\!\xi\left(1\!+\!trs\right)}{(1\!-\!\xi)\left(1\!
+\!trs\right)^2}+\left(\dfrac{1}{\delta}\right)\dfrac{1\!-\!trs\!-\!\xi\left(1\!+\!trs\right)}
{(1\!-\!\xi)\left(1+trs\right)^3}\!\right)r^{\delta/\nu-1}s^{\delta/\mu-1}\!drds\!\right)\!dt
\end{align*}}}
{\small{
\begin{align*}
\geq\int_0^1&\lambda(t)\left(\!\int_0^1\!\!\!\int_0^1\!\left(\left(1\!-\!\dfrac{1}{\delta}\right)
\dfrac{1}{\left(1\!+\!trs\right)^2}+\left(\dfrac{1}{\delta}\right)\dfrac{1\!-\!trs}
{\left(1+trs\right)^3}\right)r^{\delta/\nu-1}s^{\delta/\mu-1}drds\right.\\
&\left.-\!\int_0^1\!\!\!\int_0^1\!\left(\!\left(\!1\!-\!\dfrac{1}{\delta}\!\right)
\dfrac{1\!-\!\xi\left(1\!+\!trs\right)}{(1\!-\!\xi)\left(1\!+\!trs\right)^2}
+\left(\dfrac{1}{\delta}\right)\dfrac{1\!-\!trs\!-\!\xi\left(1\!+\!trs\right)}
{(1\!-\!\xi)\left(1+trs\right)^3}\!\right)r^{\delta/\nu-1}s^{\delta/\mu-1}\!drds\!\right)\!dt
\end{align*}}}
{\small{
\begin{align*}
=\int_0^1\!\!\!\lambda(t)\left(\!\int_0^1\!\!\!\int_0^1\!\left(\left(1\!-\!\dfrac{1}{\delta}\right)
\dfrac{\xi trs}{(1\!-\!\xi)\left(1\!+\!trs\right)^2}\!+\!\left(\dfrac{1}{\delta}\right)\dfrac{2\xi
trs}{(1\!-\!\xi)\left(1+trs\right)^3}\right)r^{\delta/\nu-1}s^{\delta/\mu-1}drds\right)\!dt
\end{align*}}}
{\small{
\begin{align*}
=\int_0^1\!\!\!\lambda(t)\left(\!\int_0^1\!\!\!\int_0^1\!\left(1+trs+\dfrac{1}{\delta}(1-trs)\right)\dfrac{\xi trs}
{(1\!-\!\xi)\left(1+trs\right)^3}r^{\delta/\nu-1}s^{\delta/\mu-1}drds\right)\!dt>0.
\end{align*}}}
Thus, ${\rm
Re}\left(z^{2-\delta}(F_\delta)^{\delta-1}(F_\delta)'\right)'>0$,
means that the function $z^{2-\delta}
(F_\delta)^{\delta-1}(F_\delta)'$ is univalent in $\mathbb{D}$.
In the next part of the theorem the following two cases are
discussed to show the order of starlikeness condition for the
function $z^{2-\delta}(F_\delta)^{\delta-1}(F_\delta)'$.
\textbf{Case (i).} Let $\gamma=0 \,(\mu=0, \nu=\alpha\geq0)$.
The function $H(z)$ defined in
\eqref{eq-generalized:starlike-H} decreases to
\begin{align*}
H(z)
=\dfrac{\alpha}{\delta} z^{1-\delta/\alpha}\left(z^{\delta/\alpha}\left(\dfrac{f}{z}\right)^\delta\right)'.
\end{align*}
Thus using duality principle, it is easy to see that
\begin{align}\label{eq-Gener:convex-f/z-gamma0}
\left(\dfrac{f}{z}\right)^\delta=\beta+\dfrac{\delta(1-\beta)}{\alpha z^{\delta/\alpha}}
\int_0^z \omega^{\delta/\alpha-1}\left(\dfrac{1+x\omega}{1+y\omega}\right)d\omega,
\end{align}
where $|x|=|y|=1$ and $z\in\mathbb{D}$. A famous result from
the theory of convolution \cite[P. 94]{Rus} states that, if
\begin{align}\label{eq-gener:Convex-equiv1}
g\in\mathcal{S}^\ast(\xi)\Longleftrightarrow
\dfrac{1}{z}(g\ast h_\xi)(z)\neq0,
\end{align}
where $h_\xi(z)$ is defined in
\eqref{eq-h(z)-extremal-starlike}.
For the function
$f(z)\!\in\!\mathcal{W}_\beta^\delta(\alpha,0)$, the generalized
integral operator $F_\delta$ defined in
\eqref{eq-weighted-integralOperator}, belongs to the class
$\mathcal{C}_\delta(\zeta)$ with the conditions
$\left(1\!-\!\frac{1}{\delta}\right)\leq\zeta\leq\left(1-\frac{1}{2\delta}\right)$
and $\delta\geq1$, is equivalent of getting
$z\left(\frac{F_\delta}{z}\right)^\delta\left(\frac{z(F_\delta)'}{F_\delta}\right)\in\mathcal{S}^\ast(\xi)$,
where $\xi$ is defined by the hypothesis,
$\xi:=1-\delta(1-\zeta)$ and $0\leq\xi\leq1/2$. Therefore,
\eqref{iff:relation:convex+starlike-integral} and
\eqref{eq-gener:Convex-equiv1} leads to
\begin{align*}
z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\in\mathcal{S}^\ast(\xi)\Longleftrightarrow
0\neq\dfrac{1}{z}\left(z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\ast
h_\xi(z)\right).
\end{align*}
Further, using logarithmic derivative of \eqref{eq-weighted-integralOperator} in the
above expression gives
\begin{align}\label{eq-convex:gener:(1)}
0\neq
&
\displaystyle\int_0^1\lambda(t)\left(\dfrac{f(tz)}{tz}\right)^\delta\left(\dfrac{tzf'(tz)}{f(tz)}\right)
dt\ast \dfrac{h_\xi(z)}{z}
\nonumber\\
=
&
\displaystyle\int_0^1\dfrac{\lambda(t)}{1-tz}dt\ast\left(\dfrac{f(z)}{z}\right)^\delta\left(\dfrac{zf'(z)}{f(z)}\right)\ast
\dfrac{h_\xi(z)}{z}.
\end{align}
Now, using a simple computation involving $z(f/z)'$, it is easy to see that $\eqref{eq-convex:gener:(1)}$ is equivalent to
\begin{align*}
0\neq&\displaystyle\int_0^1\dfrac{\lambda(t)}{1-tz}dt\ast\left(\left(1-\dfrac{1}{\delta}\right)
\left(\dfrac{f(z)}{z}\right)^\delta+\dfrac{1}{\delta}\left(z\left(\dfrac{f(z)}{z}\right)^\delta\right)'\right)\ast
\dfrac{h_\xi(z)}{z}\\
=&\displaystyle\int_0^1\lambda(t){\,}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt\ast\left(\dfrac{f(z)}{z}\right)^\delta.
\end{align*}
Substituting the value of $(f/z)^\delta$ from
\eqref{eq-Gener:convex-f/z-gamma0} will give
{\small{
\begin{align*}
0\neq\left(\displaystyle\int_0^1\lambda(t){\,}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt\right)\ast\left(\beta+\dfrac{\delta(1-\beta)}{\alpha z^{\delta/\alpha}}
\int_0^z
\omega^{\delta/\alpha-1}\left(\dfrac{1+x\omega}{1+y\omega}\right)d\omega\right)
\end{align*}}}
{\footnotesize{
\begin{align*}
\quad=(1-\beta)\left(\displaystyle\int_0^1\lambda(t)\left(\dfrac{\delta}{\alpha z^{\delta/\alpha}}
\int_0^z\omega^{\delta/\alpha-1}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(t\omega)}{t\omega}+
\dfrac{1}{\delta}h_\xi'(t\omega)\right)d\omega \right)dt+\dfrac{\beta}{1-\beta}\right)\ast\dfrac{1+xz}{1+yz}.
\end{align*}}}
Again from \cite[Pg. 23]{Rus} the above expression
is true if, and only if,
{\small{
\begin{align*}
{\rm Re}{\,}(1-\beta)\left(\displaystyle\int_0^1\lambda(t)\left(\dfrac{\delta}{\alpha z^{\delta/\alpha}}
\int_0^z\omega^{\delta/\alpha-1}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(t\omega)}{t\omega}+
\dfrac{1}{\delta}h_\xi'(t\omega)\right)d\omega \right)dt+\dfrac{\beta}{1-\beta}\right)>\dfrac{1}{2}
\end{align*}}}
or equivalently,
{\small{
\begin{align*}
{\rm Re}{\,}(1-\beta)\left(\displaystyle\int_0^1\lambda(t)\left(\dfrac{\delta}{\alpha z^{\delta/\alpha}}
\int_0^z\omega^{\delta/\alpha-1}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(t\omega)}{t\omega}+
\dfrac{1}{\delta}h_\xi'(t\omega)\right)d\omega \right)dt+\dfrac{\beta-\frac{1}{2}}{1-\beta}\right)>0.
\end{align*}}}
Using the condition on $\beta$ given in
\eqref{Beta-Cond-Generalized:Convex}, the above inequality
reduces to
\begin{align*}
{\rm Re}\displaystyle\int_0^1\lambda(t)\left(\dfrac{\delta}{\alpha z^{\delta/\alpha}}
\int_0^z\omega^{\delta/\alpha-1}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(t\omega)}{t\omega}+
\dfrac{1}{\delta}h_\xi'(t\omega)\right)d\omega -q_{0,\alpha}^\delta(t)\right)dt\geq0.
\end{align*}
Changing the variable $t\omega=u$, integrating by parts with respect to $t$ and on further
using \eqref{eq-generalized:convex-q:gamma0} and
\eqref{eq-weighted:Lambda}, the above inequality gives
\begin{align*}
{\rm Re}\displaystyle\int_0^1\Lambda_\alpha^\delta(t)\dfrac{d}{dt}\left(\dfrac{\delta}{\alpha z^{\delta/\alpha}}
\int_0^{tz}u^{\delta/\alpha-1}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(u)}{u}+
\dfrac{1}{\delta}h_\xi'(u)\right)du -t^{\delta/\alpha}q_{0,\alpha}^\delta(t)\right)dt\geq0
\end{align*}
or equivalently,
{\footnotesize{
\begin{align*}
{\rm Re}\int_0^1t^{{\delta}/{\alpha}-1}\Lambda_\alpha^\delta(t)\left[\left(1-\dfrac{1}{\delta}\right)
\left(\dfrac{h_\xi(tz)}{tz}-\dfrac{1-\xi(1+t)}{(1-\xi)(1+t)^2}\right)+\left(\dfrac{1}{\delta}\right)
\left(h_\xi'(tz)-\dfrac{1-t-\xi(1+t)}{(1-\xi)(1+t)^3}\right)\right]dt\geq0.
\end{align*}}}
\textbf{Case (ii).} Let $\gamma>0$ ($\mu>0$, $\nu>0$). Using
the conditions \eqref{iff:relation:convex+starlike-integral}
and \eqref{eq-gener:Convex-equiv1}, the integral transform
$V_\lambda^\delta(\mathcal{W}_\beta^\delta(\alpha,\gamma))\subset
\mathcal{C}_\delta(\zeta)$, for
$1-\frac{1}{\delta}\leq\zeta\leq\left(1-\frac{1}{2\delta}\right)$,
$\delta\geq1$ is equivalent of getting
\begin{align*}
0\neq\dfrac{1}{z}\left(z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\ast
h_\xi(z)\right),
\end{align*}
where $\xi=1-\delta(1-\zeta)$ and $0\leq\xi\leq1/2$. Hence
using \eqref{eq-Gener:Convex-Simplified-zf/z:Deriv} and
\eqref{eq-convex:gener:(1)}, a simple computation similar to case (i) reduces the
above expression to
\begin{align*}
0\neq\displaystyle\int_0^1\lambda(t){\,}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt\ast\left(\dfrac{f(z)}{z}\right)^\delta.
\end{align*}
Using \eqref{eq-Gener:Convex-Simplified-zf/z:Deriv} in the above inequality provides
\begin{align*}
0\neq&\displaystyle\int_0^1\lambda(t){\,}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt\ast\left[\beta+(1-\beta)\left(\dfrac{1+xz}{1+yz}\right)\right]\ast \psi_{\mu,\nu}^\delta(z)\\
=&(1-\beta)\left(\displaystyle\int_0^1\!\!\!\lambda(t)\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt+\dfrac{\beta}{1-\beta}\right)\ast
\psi_{\mu,\nu}^\delta(z)\ast\left(\dfrac{1+xz}{1+yz}\right).
\end{align*}
which is true if, and only if,
\begin{align*}
{\rm Re}{\,}(1-\beta)\left(\displaystyle\int_0^1\lambda(t){\,}\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt+\dfrac{\beta}{1-\beta}\right)\ast
\psi_{\mu,\nu}^\delta(z)>\dfrac{1}{2}
\end{align*}
or equivalently,
\begin{align*}
{\rm Re}{\,}(1-\beta)\left(\displaystyle\int_0^1\lambda(t){\,}
\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)\right){\,}dt+\dfrac{\beta-\frac{1}{2}}{1-\beta}\right)\ast
\psi_{\mu,\nu}^\delta(z)>0.
\end{align*}
Using the condition on $\beta$ given in
\eqref{Beta-Cond-Generalized:Convex}, the above inequality
becomes
\begin{align*}
{\rm Re}{\,}\displaystyle\int_0^1\lambda(t){\,}
\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+\dfrac{1}{\delta}h_\xi'(tz)-q_{\mu,\nu}^\delta(t)\right){\,}dt\ast
\psi_{\mu,\nu}^\delta(z)\geq0
\end{align*}
which on further using \eqref{eq-Gener:convex-psi:munu} leads
to
{\small{
\begin{align*}
{\rm Re}{\,}\displaystyle\int_0^1\lambda(t)
\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tz)}{tz}+
\dfrac{1}{\delta}h_\xi'(tz)-q_{\mu,\nu}^\delta(t)\right)dt\ast
\int_0^1\int_0^1\dfrac{1}{(1-u^{\nu/\delta/}v^{\mu/\delta}z)}{\,}dudv\geq0
\end{align*}}}
or equivalently,
{\small{
\begin{align*}
{\rm Re}\displaystyle\int_0^1\lambda(t)\left(\dfrac{\delta^2}{\mu\nu}
\int_0^1\int_0^1\left(\left(1-\dfrac{1}{\delta}\right)\dfrac{h_\xi(tzrs)}{tzrs}+
\dfrac{1}{\delta}h_\xi'(tzrs)\right)r^{\delta/\nu-1}s^{\delta/\mu-1}drds-q_{\mu,\nu}^\delta(t)\right)dt\geq0.
\end{align*}}}
Changing the variable $tr=\omega$,
integrating with respect to $t$ and using \eqref{eq-weighted:Lambda} leads to
{\small{
\begin{align*}
{\rm Re}{\,}\displaystyle\int_0^1\!\!\!\!\Lambda_\nu^\delta(t)\dfrac{d}{dt}\!\left(\!\dfrac{\delta^2}{\mu\nu}\!
\int_0^t\!\!\!\int_0^1\!\!\left(\!\left(\!1\!-\!\dfrac{1}{\delta}\!\right)\dfrac{h_\xi(\omega zs)}{\omega zs}\!+\!
\dfrac{1}{\delta}h_\xi'(\omega zs)\!\right)\omega^{\delta/\nu-1}s^{\delta/\mu-1}ds d\omega-t^{\delta/\nu}q_{\mu,\nu}^\delta(t)\!\right)dt\!>\!0.
\end{align*}}}
Further, using \eqref{eq-Gener:Convex-q:gamma>0} reduces the
above inequality to
\begin{align*}
{\rm Re}{\,}\displaystyle\int_0^1\Lambda_\nu^\delta(t)t^{\delta/\nu-1}\!\left(
\int_0^1\left(\left(1-\dfrac{1}{\delta}\right)\left(\dfrac{h_\xi(stz)}{stz}-\dfrac{1-\xi(1+st)}
{(1-\xi)(1+st)^2}\right)\right.\right.\\
\left.\left.+\left(\dfrac{1}{\delta}\right)\left(h_\xi'(stz)-\dfrac{1-st-\xi(1+st)}{(1-\xi)(1+st)^3}\right)\right)
s^{\delta/\mu-1}ds\right)dt\geq0.
\end{align*}
Changing the variable $ts=\eta$,
in the above expression,
integrating with respect to $t$ and using
\eqref{eq-weighted:Pi} gives
\begin{align*}
{\rm Re}{\,}\displaystyle\int_0^1\Pi_{\mu,\nu}^\delta(t)\dfrac{d}{dt}\left(
\int_0^t\left(\left(1-\dfrac{1}{\delta}\right)\left(\dfrac{h_\xi(\eta z)}{\eta z}-\dfrac{1-\xi(1+\eta)}
{(1-\xi)(1+\eta)^2}\right)\right.\right.\\
\left.\left.+\left(\dfrac{1}{\delta}\right)\left(h_\xi'(\eta z)-\dfrac{1-\eta-\xi(1+\eta)}{(1-\xi)(1+\eta)^3}\right)\right)
\eta^{\delta/\mu-1}d\eta\right)dt\geq0
\end{align*}
or equivalently,
{\small{
\begin{align*}
{\rm Re}\!\displaystyle\int_0^1\!\!\Pi_{\mu,\nu}^\delta(t) t^{\delta/\mu-1}
\left[\!\left(1\!-\!\dfrac{1}{\delta}\right)\!\!\left(\!\dfrac{h_\xi(tz)}{tz}\!-\!\dfrac{1\!-\!\xi(1\!+\!t)}{(1\!-\!\xi)(1\!+\!t)^2}\!\right)
\!+\!\left(\dfrac{1}{\delta}\right)\!\!\left(\!h_\xi'(tz)\!-\!\dfrac{1\!-\!t\!-\!\xi(1\!+\!t)}{(1\!-\!\xi)(1\!+\!t)^3}\!\right)\!\right]\!dt\!\geq\!0
\end{align*}}}
which clearly implies that the function
$\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)\geq0$ and the proof
is complete.
Now, to validate the condition of sharpness for the function
$f(z)\in\mathcal{W}_\beta^\delta(\alpha,\gamma)$, satisfying
the differential equation
\begin{align}\label{eq-generalized-f/z-extremal}
\dfrac{\mu\nu}{\delta^2}{\,} z^{1-\delta/\mu}\left(z^{\delta/\mu-\delta/\nu+1}
\left( z^{\delta/\nu}\left(\dfrac{f}{z}\right)^\delta\right)'\right)'
=\beta+(1-\beta)\dfrac{1+z}{1-z}
\end{align}
with the parameter $\beta<1$ defined in
\eqref{Beta-Cond-Generalized:Convex}. From
\eqref{eq-generalized-f/z-extremal}, a simple calculation gives
\begin{align}\label{eq-Gener:Convex-Shapness}
\left(\dfrac{f}{z}\right)^\delta
=1+2(1-\beta)\sum_{n=1}^\infty \dfrac{\delta^2z^n}{(\delta+n\nu)(\delta+n\mu)}.
\end{align}
Substituting \eqref{eq-Gener:Convex-Shapness} in
\eqref{eq-Gener:Convex-Simplified-zf/z:Deriv} will give
\begin{align}\label{eq-generalized-convex-simpli-f/z-1}
z\left(\dfrac{f}{z}\right)^\delta\left(\dfrac{zf'}{f}\right)
=z+2(1-\beta)\sum_{n=1}^{\infty}\dfrac{(n+\delta)\delta z^{n+1}}{(\delta+n\nu)(\delta+n\mu)}.
\end{align}
Further, substituting
\eqref{eq-generalized-convex-simpli-f/z-1} in the expression involving the logarithmic derivative of \eqref{eq-weighted-integralOperator} leads to
\begin{align}\label{eq-Gener:Convex-F:series}
z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)
&=\int_0^1\dfrac{\lambda(t)}{t}tz\left(\dfrac{f(tz)}{tz}\right)^\delta\left(\dfrac{tzf'(tz)}{f(tz)}\right)dt\nonumber\\
&=z+2(1-\beta)\sum_{n=1}^{\infty}\dfrac{(n+\delta)\delta\tau_n z^{n+1}}{(\delta+n\nu)(\delta+n\mu)}
\end{align}
where $\tau_n=\int_0^1t^n\lambda(t)dt$. Differentiating
\eqref{eq-Gener:Convex-F:series} will give
\begin{align*}
\left(z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\right)'
=1+2(1-\beta)\sum_{n=1}^{\infty}\dfrac{(n+1)(n+\delta)\delta\tau_n z^{n}}{(\delta+n\nu)(\delta+n\mu)}.
\end{align*}
which clearly implies
\newline
$\displaystyle
z\left.\left(z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\right)'\right|_{z=-1}
$
\begin{align}\label{eq-gener-Convex:Mainthm:(1)}
=-1-2(1-\beta)\sum_{n=1}^{\infty}\dfrac{(-1)^{n}(n+1-\xi)(n+\delta)\delta\tau_n }{(\delta+n\nu)(\delta+n\mu)}
\quad+2(1-\beta)\xi\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(n+\delta)\delta\tau_n }{(\delta+n\nu)(\delta+n\mu)}.
\end{align}
The series expansion of the function $q_{\mu,\nu}^\delta(t)$
defined in \eqref{eq-Gener:Convex-Integral:q:gamma>0} is
\begin{align}\label{eq-Gener:Convex-q:series}
q_{\mu,\nu}^\delta(t)=1+\dfrac{\delta}{(1-\xi)}\sum_{n=1}^\infty\dfrac{(n+\delta)(n+1-\xi)(-1)^nt^n}{(\delta+n\nu)(\delta+n\mu)}.
\end{align}
whose representation in the form of generalized hypergeometric
function is given as
\begin{align}\label{eq-gener:convex-q:hyper:series}
q_{\mu,\nu}^\delta(t)=\,_5F_4\left(1,(1+\delta),(2-\xi),\dfrac{\delta}{\mu},\dfrac{\delta}{\nu};{\,}
\delta,(1-\xi),\left(1+\dfrac{\delta}{\mu}\right),\left(1+\dfrac{\delta}{\nu}\right);{\,}-t\right).
\end{align}
Using \eqref{eq-Gener:Convex-q:series} in
\eqref{Beta-Cond-Generalized:Convex} gives
\begin{align}\label{eq:Gener:Convex-beta-series}
\dfrac{\left(\beta-\frac{1}{2}\right)}{(1-\beta)}
=-1-\dfrac{\delta}{(1-\xi)}\sum_{n=1}^\infty\dfrac{(n+\delta)(n+1-\xi)(-1)^n\tau_n}{(\delta+n\nu)(\delta+n\mu)}.
\end{align}
From \eqref{eq-Gener:Convex-F:series} and
\eqref{eq:Gener:Convex-beta-series}, the expression
\eqref{eq-gener-Convex:Mainthm:(1)} is equivalent to
\begin{align*}
z\left.\left(z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\right)'\right|_{z=-1}
=\xi {\,}\left.z\left(\dfrac{F_\delta}{z}\right)^\delta\left(\dfrac{z(F_\delta)'}{F_\delta}\right)\right|_{z=-1},
\end{align*}
which means that the result is sharp.
\end{proof}
\begin{remark}
\begin{enumerate}[1.]
\item For $\delta=1$ and $\xi=0$, {\rm Theorem
\ref{Thm-Gener:Convex-Mainresult}} is similar to
{\rm\cite[Theorem 3.1]{MahnazC}}.
\item For $\delta=1$, {\rm Theorem
\ref{Thm-Gener:Convex-Mainresult}} reduces to
{\rm\cite[Theorem 3.1]{SarikaC}}.
\end{enumerate}
\end{remark}
The condition $\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)\geq 0$
derived in Theorem \ref{Thm-Gener:Convex-Mainresult} is
difficult to use, therefore a simpler sufficient condition is
presented in the next result.
\begin{theorem}\label{Thm:Main-Gener:Convex-Decreas}
Let $\mu\in[1/2,1]$, $\nu\geq1$ and
$\left(1-\frac{1}{\delta}\right)\leq\zeta\leq
\left(1-\frac{1}{2\delta}\right)$, where $\delta\geq1$. Let
$\beta<1$ satisfy \eqref{Beta-Cond-Generalized:Convex} and
\begin{align}\label{eq-Gener:Convex-Decre-Cond}
\dfrac{t^{{1}/{\mu}(\delta-1)}\left(\delta\left(1-\frac{1}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)-
t\left(\Pi_{\mu,\nu}^{\delta}(t)\right)^\prime\right)}{(\log(1/t))^{3-2\delta(1-\zeta)}}
\end{align}
is decreasing on $(0,1)$. Then the function
$\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)(z)\geq0$, where
$\xi=1-\delta(1-\zeta)$ and $0\leq\xi\leq1/2$.
\end{theorem}
\begin{proof}
Since the function
\begin{align*}
\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)(z)=&\int_0^1t^{{\delta}/{\mu}-1}\Pi_{\mu,\nu}^\delta(t)
\left(\left(1-\dfrac{1}{\delta}\right)\left({\rm Re}\dfrac{h_\xi(tz)}{tz}+\dfrac{1-\xi(1+t)}{(1-\xi)(1+t)^2}\right)\right.\\
&\quad+\left.\left(\dfrac{1}{\delta}\right)
\left({\rm Re}{\,}h_\xi'(tz)-\dfrac{1-t-\xi(1+t)}{(1-\xi)(1+t)^3}\right)\right)dt,
\end{align*}
where $\xi=1-\delta(1-\zeta)$ and $0\leq\xi\leq1/2$.
Equivalently, it can also be written as
\begin{align*}
\mathcal{M}_{\Pi_{\mu,\nu}^\delta}(h_\xi)(z)=&\int_0^1 t^{{\delta}/{\mu}-1}\Pi_{\mu,\nu}^\delta(t)
\left(\left(1-\dfrac{1}{\delta}\right)\left({\rm Re}\dfrac{h_\xi(tz)}{tz}+\dfrac{1-\xi(1+t)}{(1-\xi)(1+t)^2}\right)\right.\\
&\left.\quad+\dfrac{d}{dt}\left(\dfrac{1}{\delta}\left({\rm Re}\dfrac{h_\xi(tz)}{z}-\dfrac{t(1-\xi(1+t))}{(1-\xi)(1+t)^2}\right)\right)\right)dt,
\end{align*}
which on further simplification gives
{\small{
\begin{align}\label{eq-Gener:Convex-Main-ineq}
\mathcal{M}_{\Pi_{\mu,\nu}^\delta}&(h_\xi)(z)\!=\!\left(\!1\!-\!\dfrac{1}{\delta}\!\right)\!\int_0^1\!
t^{{\delta}/{\mu}-1}\Pi_{\mu,\nu}^\delta(t)\left({\rm
Re}\dfrac{h_\xi(tz)}{tz}
\!+\!\dfrac{1\!-\!\xi(1\!+\!t)}{(1\!-\!\xi)(1\!+\!t)^2}\right)\!dt\nonumber\\
&\quad\quad+\!\int_0^1\!t^{{\delta}/{\mu}-1}\left(\dfrac{1}{\delta}\right)\left(\left(1\!-\!\dfrac{\delta}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)\!
-\!t\left(\Pi_{\mu,\nu}^\delta(t)\right)'\right)\!\!\left(\!{\rm
Re}\dfrac{h_\xi(tz)}{tz}
\!+\!\dfrac{1\!-\!\xi(1\!+\!t)}{(1\!-\!\xi)(1\!+\!t)^2}\right)\!dt\nonumber\\
&=\int_0^1 t^{{\delta}/{\mu}-1}\left(\left(1\!-\!\frac{1}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)\!-\!\left(\dfrac{1}{\delta}\right)
t\left(\Pi_{\mu,\nu}^{\delta}(t)\right)^\prime\right)
\left({\rm Re}\dfrac{h_\xi(tz)}{tz}-\dfrac{1-\xi(1+t)}{(1\!-\!\xi)(1+t)^2}\right)dt.
\end{align}}}
The right side of \eqref{eq-Gener:Convex-Main-ineq} is bounded
from below. So, due to the existence of lower bound, the
minimum principle states that, the minimum value of
\eqref{eq-Gener:Convex-Main-ineq} lies on the boundary i.e., on
$|z|=1$, where $z\neq1$. Now, minimizing ${\rm
Re}(h_\xi(tz)/(tz))$ with respect to $\epsilon$ will give
\begin{align*}
{\rm Re}\dfrac{h_\xi(tz)}{tz}\geq \dfrac{1}{2(1-\xi)}\left({\rm Re}\dfrac{2(1-\xi)+(2\xi-1)tz}{(1-tz)^2}-\dfrac{t}{|1-tz|^2}\right).
\end{align*}
Hence, \eqref{eq-Gener:Convex-Main-ineq} is equivalent of
obtaining
\begin{align*}
\int_0^1 t^{{\delta}/{\mu}-1}&\left(\delta\left(1-\dfrac{1}{\mu}\right)
\Pi_{\mu, \,\nu}^\delta(t)-t\left(\Pi_{\mu, \,\nu}^\delta(t)\right)^\prime\right)\\
&\left({\rm Re}\dfrac{2(1-\xi)+(2\xi-1)tz}{(1-tz)^2}
-\dfrac{t}{|1-tz|^2}-\dfrac{2(1-\xi(1+t))}{(1+t)^2}\right)dt\geq0.
\end{align*}
The equality of the above integral exist at $z=-1$. Since
$|z|=1$ and $z\neq1$, now letting ${\rm Re}z=y$ will reduce it
to considering
\begin{align*}
H_\Pi^{(\xi)}(y)\!=\!\int_0^1\! &t^{{\delta}/{\mu}-1}\left(\delta\left(1-\dfrac{1}{\mu}\right)
\Pi_{\mu, \,\nu}^\delta(t)-t\left(\Pi_{\mu, \,\nu}^\delta(t)\right)^\prime\right)\\
&\left(\!\dfrac{t(3\!-\!4(1\!\!+\!y)t\!+\!2(4y\!-\!1)t^2\!+\!4(y\!-\!1)t^3\!-\!t^4)}
{(1-2yt+t^2)^2(1+t)^2}\!-\!\dfrac{2\xi(1-t)}{(1\!-\!2yt\!+\!t^2)(1\!\!+\!t)}\!\right)\!dt\!\geq\!0
\end{align*}
where $|z|=1$ and $z\neq1$, gives $-1\leq y<1$. Since the term
$(1+y)\geq0$, $H_\Pi^{(\xi)}(y)$ can be written in
the series form as
\begin{align*}
H_\Pi^{(\xi)}(y)=\sum_{j=0}^\infty H_{j,\Pi}^{(\xi)}(1+y)^j,\quad\quad |1+y|<2.
\end{align*}
An easy computation shows that the $j$th term of
$H_{j,\Pi}^{(\xi)}$ is a positive multiple of
\begin{align*}
\tilde{H}_{j,\Pi}^{(\xi)}=\int_0^1t^{{\delta}/{\mu}-1}\left(\delta\left(1-\dfrac{1}{\mu}\right)
\Pi_{\mu, \,\nu}^\delta(t)-t\left(\Pi_{\mu, \,\nu}^\delta(t)\right)^\prime\right)(s_j(t)-2\xi u_j(t))dt,
\end{align*}
where
\begin{align*}
s_j(t):=\dfrac{(j+3)t^{j+1}}{(1+t)^{2j+4}}\left(1-2t+\dfrac{j-1}{j+3}t^2\right)\quad\quad{\rm and} \quad
u_j(t):=\dfrac{t^{j+1}}{(1+t)^{2j+4}}(1-t^2).
\end{align*}
to give
\begin{align*}
s_j(t)-2\xi u_j(t)=\dfrac{t^{j+1}}{(1+t)^{2j+4}}\, v(t),
\end{align*}
with $v(t):=\left((j+3)(1-2t)+(j-1)t^2-2\xi(1-t^2)\right)$.
The function $v(t)$ is decreasing on $t\in(0,1)$. At $t=0$,
$v(t)$ is positive and at $t=1$, $v(t)$ is negative, which
clearly implies that the function $(s_j(t)-2\xi u_j(t))$ has
exactly one zero for $t\in(0,1)$. Set this zero by
$t_{j}^{(\xi)}$. Therefore, $(s_j(t)-2\xi u_j(t))>0$, for
$0\leq t<t_{j}^{(\xi)}$ and $(s_j(t)-2\xi u_j(t))<0$, for
$t_{j}^{(\xi)}<t<1$.
Now, define the functions
\begin{align}\label{eq-decre-H_j1}
\tilde{H}_{j}^{(\xi)}=\int_0^1 t^{{1}/{\mu}-1}(s_j(t)-2\xi u_j(t))\left(\log\left(\frac{1}{t}\right)\right)^{1+2\xi}dt
\end{align}
and
\begin{align*}
\tilde{\Pi}_{\mu,\nu}^{\delta,\xi}(t)=&t^{\frac{1}{\mu}(\delta-1)}\left(\delta\left(1-\frac{1}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)-
t\left(\Pi_{\mu,\nu}^{\delta}(t)\right)^\prime\right)\\
&-\dfrac{(t_j^{(\xi)})^{\frac{1}{\mu}(\delta-1)}\left(\delta\left(\!1\!-\!\dfrac{1}{\mu}\!\right)\! \Pi_{\mu,
\,\nu}^\delta(t_j^{(\xi)})- t_j^{(\xi)}\left(\Pi_{\mu, \,\nu}^\delta(t_j^{(\xi)})\right)^\prime\right)}
{(\log(1/t_{j}^{(\xi)}))^{1+2\xi}}(\log(1/t))^{1+2\xi}.
\end{align*}
Since the hypothesis \eqref{eq-Gener:Convex-Decre-Cond} of the
theorem implies that the function
\begin{align*}
\dfrac{t^{\frac{1}{\mu}(\delta-1)}\left(\delta\left(1-\frac{1}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)-
t\left(\Pi_{\mu,\nu}^{\delta}(t)\right)^\prime\right)}{(\log(1/t))^{1+2\xi}}
\end{align*}
is decreasing, where $\xi=1-\delta(1-\zeta)$ and
$0\leq\xi\leq1/2$, thus it is easy to observe that the
condition on $(s_j(t)-2\xi u_j(t))$ and the function
$\tilde{\Pi}_{\mu,\nu}^{\delta,\xi}(t)$ have same sign for
$t\in(0,1)$. Hence
\begin{align}\label{eq-decre-H_j2}
0&\leq\int_0^1 t^{\frac{1}{\mu}-1}\tilde{\Pi}_{\mu,\nu}^{\delta,\xi}(t)(s_j(t)-2\xi u_j(t))dt\nonumber \\
&=\tilde{H}_{j,\Pi}^{(\xi)}-\dfrac{(t_j^{(\xi)})^{\frac{1}{\mu}(\delta-1)}\left(\delta\left(1-\frac{1}{\mu}\right)\Pi_{\mu,
\,\nu}^\delta(t_j^{(\xi)})- t_j^{(\xi)}\left(\Pi_{\mu, \,\nu}^\delta(t_j^{(\xi)})\right)^\prime\right)}
{(\log(1/t_{j}^{(\xi)}))^{1+2\xi}}
\tilde{H}_{j}^{(\xi)}.
\end{align}
Using
\eqref{eq-weighted:Lambda} and \eqref{eq-weighted:Pi}, we have
\begin{align*}
\left(\Lambda_\nu^\delta(t)\right)'=-\lambda(t)t^{-\delta/\nu}\quad{\rm and}\quad
\left(\Pi_{\mu,\nu}^\delta(t)\right)'=-\Lambda_\nu^\delta(t)t^{-\delta/\mu+\delta/\nu-1}
\end{align*}
which clearly shows that
\begin{align*}
\dfrac{d}{dt}\left(\delta\left(1-\dfrac{1}{\mu}\right) \Pi_{\mu,\,\nu}^\delta(t)- t\left(\Pi_{\mu, \,\nu}^\delta(t)\right)^\prime\right)
&=\dfrac{d}{dt}\left(\delta\left(1-\dfrac{1}{\mu}\right) \Pi_{\mu,\,\nu}^\delta(t)+
t^{\delta/\nu-\delta/\mu}\Lambda_{\nu}^\delta(t)\right)\\
&=-\delta\left(1-\dfrac{1}{\nu}\right)t^{\delta/\nu-\delta/\mu-1}\Lambda_{\nu}^\delta(t)-t^{-\delta/\mu}\lambda(t)<0.
\end{align*}
for $\nu\geq1$ and $t\in(0,1)$. Thus, the above condition
implies
\begin{align*}
\delta\left(1-\dfrac{1}{\mu}\right) \Pi_{\mu,\,\nu}^\delta(t)- t\left(\Pi_{\mu, \,\nu}^\delta(t)\right)^\prime>0.
\end{align*}
Using similar arguments as in \cite[Page 280]{Aghalary} for the positivity of $\tilde{H}_{j}^{(\xi)}$ defined by
\eqref{eq-decre-H_j1}, from \eqref{eq-decre-H_j2}, it follows that
$\tilde{H}_{j,\Pi}^{(\xi)}\geq0$ and this completes the proof.
\end{proof}
\section{Applications of theorem \ref{Thm:Main-Gener:Convex-Decreas}}\label{Sec-Gener:Convex-Suff:Cond}
To apply Theorem \ref{Thm:Main-Gener:Convex-Decreas}, for the
case $\gamma>0$ $(\mu>0, \nu>0)$, it is required to show that
the function
\begin{align*}
\dfrac{t^{{(\delta-1)}/{\mu}}\left(\delta\left(1-{1}/{\mu}\right)\Pi_{\mu,\nu}^\delta(t)-
t\left(\Pi_{\mu,\nu}^{\delta}(t)\right)^\prime\right)}{(\log(1/t))^{3-2\delta(1-\zeta)}}
\end{align*}
is decreasing in the range $t\in(0,1)$, where $\mu\in[1/2,1]$,
$\nu\geq1$, $\delta\geq1$ and
$\left(1-\frac{1}{\delta}\right)\leq\zeta\leq
\left(1-\frac{1}{2\delta}\right)$. Since
$\xi=(1-\delta(1-\zeta))$, thus using \eqref{eq-weighted:Pi},
the above expression can be rewritten as
\begin{align*}
g(t):=\dfrac{\delta\left(1-\frac{1}{\mu}\right)t^{{\delta}/{\mu}-1/\mu}\,\Pi_{\mu,\nu}^\delta(t)+
t^{\delta/\nu-1/\mu}\,\Lambda_{\nu}^{\delta}(t)}{(\log(1/t))^{1+2\xi}},
\end{align*}
where $\xi\in[0,1/2]$. Note that the chosen function
$\lambda(t)$ satisfy the condition $\lambda(1)=0$. Therefore,
in the overall discussion, the assumed conditions hold.
Taking the derivative of $g(t)$ and using
\eqref{eq-weighted:Lambda} and \eqref{eq-weighted:Pi} will give
\begin{align*}
g'(t)=\dfrac{t^{{\delta}/{\mu}-1/\mu-1}h(t)}{\left(\log\frac{1}{t}\right)^{2(1+\xi)}}&
\left[\delta\left(1-\frac{1}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)+
\left(1+\delta\left(\frac{1}{\nu}-1\right)\dfrac{\log\frac{1}{t}}{h(t)}\right)\,t^{\delta/\nu-\delta/\mu}\Lambda_{\nu}^{\delta}(t)\right.\\
&\left.-t^{1-\delta/\mu}\frac{\log\frac{1}{t}}{h(t)}\lambda(t)\right],
\end{align*}
where the function
$h(t):=\frac{1}{\mu}(\delta-1)\log\frac{1}{t}+(1+2\xi)$, which
by simple computation for $0<t<1$, $\delta\geq1$ and
$0\leq\xi\leq1/2$ gives $h(t)\geq1$.
Therefore, proving $g'(t)\leq0$ is equivalent of getting
$k(t)\leq0$, where
\begin{align*}
k(t):=\delta\left(1-\frac{1}{\mu}\right)\Pi_{\mu,\nu}^\delta(t)+
\left(1+\delta\left(\frac{1}{\nu}-1\right)\dfrac{\log\frac{1}{t}}{h(t)}\right)t^{\delta/\nu-\delta/\mu}\,\Lambda_{\nu}^{\delta}(t)
-t^{1-\delta/\mu}\frac{\log\frac{1}{t}}{h(t)}\lambda(t).
\end{align*}
Clearly $k(1)=0$ implies that if $k(t)$ is increasing function
of $t\in(0,1)$ then $g'(t)\leq0$. Hence, it is required to show
that
\begin{align*}
k'(t)=t^{\delta/\nu-{\delta}/{\mu}-1}\dfrac{l(t)}{h(t)},
\end{align*}
where
\begin{align*}
l(t):=&\left(\frac{\delta}{\nu}-\delta\right)\Lambda_{\nu}^{\delta}(t)
\left[\left(\frac{\delta}{\nu}-\frac{1}{\mu}\right)\log\frac{1}{t}+1+2\xi-
\dfrac{(1+2\xi)}{h(t)}\right]\\
&+t^{1-\delta/\nu}\lambda(t)\left[\left(\frac{1}{\mu}-\frac{\delta}{\nu}+\delta-1\right)\log\frac{1}{t}-1-2\xi
+\dfrac{(1+2\xi)}{h(t)}\right]-t^{2-\delta/\nu}\log\frac{1}{t}\lambda'(t)\geq0.
\end{align*}
Now, using the hypothesis $\lambda(1)=0$ implies that $l(1)=0$.
Therefore $l(t)$ is decreasing function of $t\in(0,1)$, i.e.,
if $l'(t)\leq0$, clearly means that the function $g(t)$ is
decreasing. Now, we calculate
{\small{
\begin{align*}
l'(t)=&\delta\left(1-\frac{1}{\nu}\right)\frac{\Lambda_{\nu}^{\delta}(t)}{t}
\left[\left(\frac{\delta}{\nu}-\frac{1}{\mu}\right)+\left(\frac{\delta}{\mu}-\frac{1}{\mu}\right)\dfrac{(1+2\xi)}{(h(t))^2}\right]
+t^{-\delta/\nu}\lambda(t)\left[(\delta-1)\left(1-\frac{1}{\mu}\right)\log\frac{1}{t}\right.\\
&\left.+\left(\frac{\delta}{\nu}-\frac{1}{\mu}+2\xi(\delta-1)\right)-\frac{(\delta-1)(1+2\xi)}{h(t)}
+\left(\frac{\delta}{\mu}-\frac{1}{\mu}\right)\frac{(1+2\xi)}{(h(t))^2}\right]\nonumber\\
&+t^{1-\delta/\nu}\lambda'(t)\left[\left(\frac{1}{\mu}+\delta-3\right)\log\frac{1}{t}-2\xi+\frac{(1+2\xi)}
{h(t)}\right]-\log\frac{1}{t}t^{2-\delta/\nu}\lambda''(t).
\end{align*}}}
Thus, the function $g'(t)\leq0$ is counterpart of the following
inequalities:
\begin{align}\label{eq-Gener:convex-Main:equiv(Cond1)}
\Lambda_{\nu}^{\delta}(t)\left[\left(\frac{1}{\mu}-\frac{\delta}{\nu}\right)(h(t))^2-(1+2\xi)\left(\frac{\delta}{\mu}-\frac{1}{\mu}\right)\right]\geq0
\end{align}
and
{\small{
\begin{align}\label{eq-Gener:convex-Main:equiv(Cond2)}
\lambda(t)\left[\left(\frac{\delta}{\nu}-\frac{1}{\mu}+2\xi(\delta-1)\right)+
(\delta-1)\left(1-\frac{1}{\mu}\right)\log\frac{1}{t}-\frac{(\delta-1)(1+2\xi)}{h(t)}
+\left(\frac{\delta}{\mu}-\frac{1}{\mu}\right)\frac{(1+2\xi)}{(h(t))^2}\right]\nonumber\\
+t\lambda'(t)\left[\left(\frac{1}{\mu}+\delta-3\right)\log\frac{1}{t}-2\xi+\frac{(1+2\xi)}
{h(t)}\right]-\log\frac{1}{t}t^{2}\lambda''(t)\leq0,
\end{align}}}
\hspace{-.10cm}for $\nu\geq1$ and $t\in(0,1)$. Letting
${(2-\delta)}/{\mu}\geq{\delta}/{\nu}$ implies that the
inequality \eqref{eq-Gener:convex-Main:equiv(Cond1)} is true,
which clearly means that the function $g(t)$ is decreasing, if
the inequality \eqref{eq-Gener:convex-Main:equiv(Cond2)} holds
along with the condition
${(2-\delta)}/{\mu}\geq{\delta}/{\nu}$, for $1\leq\delta<2$,
$\mu\in[1/2,1]$ and $\nu\geq1$.
The function $h(t)\geq1$ and
$\left(1-\delta/\mu\right)+\left(\delta/\mu-1/\mu\right)/h(t)\leq0$,
for $1/2\leq\mu\leq1$ and $\delta\geq1$. Thus the inequality
\eqref{eq-Gener:convex-Main:equiv(Cond2)} is true when
\begin{align}\label{eq-Gener:convex-Main:equiv(Cond3)}
\lambda(t)\left[\left(\frac{1}{\mu}-\frac{\delta}{\nu}-2\xi(\delta-1)\right)+
(\delta-1)\left(\frac{1}{\mu}-1\right)\log\frac{1}{t}+\left(\delta-\dfrac{\delta}{\mu}\right)\frac{(1+2\xi)}{h(t)}\right]\nonumber\\
+t\lambda'(t)\left[\left(3-\delta-\frac{1}{\mu}\right)\log\frac{1}{t}+2\xi-\frac{(1+2\xi)}{h(t)}\right]+\log\frac{1}{t}t^{2}\lambda''(t)\geq0.
\end{align}
In order to use the above condition for the application purposes, we consider the following.
For the parameters $A,B,C>0$, set
\begin{align}\label{eq-Generalized-hypergeometric-fn}
\lambda(t)=K t^{B-1}(1-t)^{C-A-B}\omega(1-t),
\end{align}
where the function
\begin{align*}
\omega(1-t)=1+\displaystyle\sum_{n=1}^\infty x_n(1-t)^n,\quad{\rm with}
\quad x_n\geq0,\quad t\in(0,1).
\end{align*}
The constant $K$ is chosen such that it satisfies normalization
condition $\int_0^1\lambda(t)dt=1$ and $(C-A-B)>0$ which
clearly implies that the function $\lambda(t)$ is zero at
$t=1$.
By an easy calculation, we get
{\small{
\begin{align}\label{eq-Generalized-hypergeometric-fn-lambda'}
\lambda'(t)=Kt^{B-2}(1-t)^{C-A-B-1}\left[\left(\frac{}{}(B-1)(1-t)-(C-A-B)t\right)\omega(1-t)-t(1-t)\omega'(1-t)\right],
\end{align}}}
and
{\small{
\begin{align}\label{eq-Generalized-hypergeometric-fn-lambda''}
\lambda^{\prime\prime}(t)=&Kt^{B-3}(1-t)^{C-A-B-2}\left[\left(\dfrac{}{}(B-1)(B-2)(1-t)^2\right.\right.\nonumber\\
&\left.\dfrac{}{}-2(B-1)(C-A-B)t(1-t)+(C-A-B)(C-A-B-1)t^2\right)\omega(1-t)\nonumber\\
&\left.+\left(\dfrac{}{}2(C-A-B)t-2(B-1)(1-t)\right)t(1-t)\omega'(1-t)+t^2(1-t)^2\omega''(1-t)\right].
\end{align}}}
Now, substituting the values of $\lambda(t)$, $\lambda'(t)$ and
$\lambda''(t)$ given in
\eqref{eq-Generalized-hypergeometric-fn},
\eqref{eq-Generalized-hypergeometric-fn-lambda'} and
\eqref{eq-Generalized-hypergeometric-fn-lambda''}, respectively
in inequality \eqref{eq-Gener:convex-Main:equiv(Cond3)} will
give the corresponding condition as
\begin{align}\label{eq-Gener:convex-Main-Gener:hyper-gamma>0}
t^2(1-t)^2\log\dfrac{1}{t}\omega''(1-t)+t(1-t)X_1(t)\omega'(1-t)+X_2(t)\omega(1-t)\geq0
\end{align}
where
{\small{
\begin{align*}
X_1(t):=\log\dfrac{1}{t}\left[(1-t)\left(\dfrac{1}{\mu}+\delta-2B-1\right)+2(C-A-B)t\right]
+(1-t)\left(-2\xi+\dfrac{(1+2\xi)}{h(t)}\right).
\end{align*}}}
and
{\footnotesize{
\begin{align*}
X_2(t):=&\log\dfrac{1}{t}\left[(1-t)^2\left[(\delta-1)\left(\frac{1}{\mu}-1\right)+(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)\right]
+(C-A-B)t\times\right.\\
&\left.\left[(1-t)\left(\dfrac{1}{\mu}+\delta-2B-1\right)+(C-A-B-1)t\right]\right]+(1-t)\left[(1-t)
\left[\left(\frac{1}{\mu}\!-\!\frac{\delta}{\nu}\!-\!2\xi(\delta\!-\!B)\right)\right.\right.\\
&\left.\left.+\left(\delta+1-B-\frac{\delta}{\mu}\right)\dfrac{(1+2\xi)}{h(t)}\right]+(C-A-B)t
\left[-2\xi+\dfrac{(1+2\xi)}{h(t)}\right]\right].
\end{align*}}}
Since the function $\omega(1-t)=1+\sum_{n=1}^\infty
x_n(1-t)^n$, with the condition $x_n\geq0$, which clearly means
that the function $\omega(1-t)$, $\omega'(1-t)$ and
$\omega''(1-t)$ are non-negative for all values of $t\in(0,1)$.
Therefore, proving inequality
\eqref{eq-Gener:convex-Main-Gener:hyper-gamma>0}, it suffice to
show
\begin{align*}
X_1(t)\geq0\quad{\rm and }\quad X_2(t)\geq0.
\end{align*}
Now, in this respect the following two cases are examined:
\noindent {\bf{Case (i)}} Let $0<B\leq\delta$. By a simple adjustment, it can be easily obtained that the
inequality $X_1(t)\geq0$ holds true if
\begin{align*}
\log\dfrac{1}{t}\left[(1-t)\left(\dfrac{1}{\mu}+\delta-2B-1\right)+2(C-A-B)t\right]
\geq2\xi(1-t),
\end{align*}
where $\xi=1-\delta(1-\zeta)$, for
$\left(1-\frac{1}{\delta}\right)\leq\zeta\leq
\left(1-\frac{1}{2\delta}\right)$. Since the right side of
the above inequality is positive for $\xi\in[0,1/2]$ and
$t\in(0,1)$, hence on using the condition
\begin{align}\label{eq-general:cond-log-1-t}
(1-t)\leq\dfrac{(1+t)}{2}\log\dfrac{1}{t},\quad t\in(0,1),
\end{align}
it is enough to get
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma>0(2)}
\left(\dfrac{1}{\mu}+\delta-1-2B-\xi\right)(1-t)+2(C-A-B-\xi)t\geq0.
\end{align}
Further, the equivalent condition for $X_2(t)\geq0$ is
obtained. By the assumed hypothesis
${(2-\delta)}/{\mu}\geq{\delta}/{\nu}$ directly implies
$1/\mu\geq\delta/\nu$. Now using this condition, the
function $X_2(t)\geq0$ is valid if
{\small{
\begin{align*}
&\log\dfrac{1}{t}\left((C-A-B)t\left[(1-t)\left(\dfrac{1}{\mu}+\delta-2B-1\right)+(C-A-B-1)t\right]\right.\\
&\left.+(1-t)^2\left[(\delta-1)\left(\frac{1}{\mu}-1\right)+(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)\right]\right)
+(1-t)\left(\dfrac{}{}(1-t)\times\right.\\
&\left.\left[-2\xi(\delta-B)-\left(\frac{1}{\mu}+B-1\right)\dfrac{(1+2\xi)}{h(t)}\right]-2\xi(C-A-B)t\right)\geq0.
\end{align*}}}
or equivalently,
{\small{
\begin{align}\label{eq-Gener:convex-Main-Gener:hyper-gamma>0-1}
&\log\dfrac{1}{t}\left((C-A-B)t\left[(1-t)\left(\dfrac{1}{\mu}+\delta-2B-1\right)+(C-A-B-1)t\right]\right.\nonumber\\
&\quad\left.+(1-t)^2\left[(\delta-1)\left(\frac{1}{\mu}-1\right)+(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)\right]\right)\nonumber\\
\geq&(1-t)\left(2\xi(C-A-B)t+(1-t)\left[2\xi(\delta-B)+\left(\frac{1}{\mu}+B-1\right)\dfrac{(1+2\xi)}{h(t)}\right]\right).
\end{align}}}
As $0\leq B\leq\delta$, therefore using the conditions
$0\leq\xi\leq1/2$, $1/2\leq\mu\leq1$, and $(C-A-B)>0$, it
is easy to check that the coefficient of $(1-t)$ on right
side of the above expression is positive. Therefore, in
view of the inequality \eqref{eq-general:cond-log-1-t}, the
condition
\eqref{eq-Gener:convex-Main-Gener:hyper-gamma>0-1} holds
true for $t\in(0,1)$ if
\begin{align*}
&2\left((C-A-B)t\left[(1-t)\left(\dfrac{1}{\mu}+\delta-2B-1\right)+(C-A-B-1)t\right]\right.\nonumber\\
&\quad\left.+(1-t)^2\left[(\delta-1)\left(\frac{1}{\mu}-1\right)+(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)\right]\right)\nonumber\\
\geq&(1+t)\left(2\xi(C-A-B)t+(1-t)\left[2\xi(\delta-B)+\left(\frac{1}{\mu}+B-1\right)\dfrac{(1+2\xi)}{h(t)}\right]\right)
\end{align*}
or equivalently,
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma>0(1)}
(1-t)^2\left[2(\delta-1)\left(\frac{1}{\mu}-1\right)+2(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)+R(t)\right]\nonumber\\
+2t(1-t)\left[(C-A-B)\left(\dfrac{1}{\mu}+\delta-2B-1-\xi\right)+R(t)\right]\nonumber\\
+2t^2(C-A-B)(C-A-B-1-2\xi)\geq0,
\end{align}
where
\begin{align*}
R(t):=\left(-\frac{1}{\mu}+1-B\right)\dfrac{(1+2\xi)}{h(t)}-2\xi(\delta-B).
\end{align*}
Consequently, the condition
\eqref{eq-Gener:convex-Main:ineq-gamma>0(1)} holds good if
the coefficients of $t^2$, $t(1-t)$, and $(1-t)^2$ are
positive. Now it remains to prove the following
inequalities:
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma>0(11)}
(C-A-B)(C-A-B-1-2\xi)\geq0,
\end{align}
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma>0(12)}
(C-A-B)\left(\dfrac{1}{\mu}+\delta-2B-1-\xi\right)+R(t)\geq0,
\end{align}
and
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma>0(13)}
2(\delta-1)\left(\frac{1}{\mu}-1\right)+2(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)+R(t)\geq0,
\end{align}
where $\xi=1-\delta+\delta\zeta$,
$\left(1-\frac{1}{\delta}\right)\leq\zeta\leq
\left(1-\frac{1}{2\delta}\right)$ and $\delta\geq1$.
\noindent {\bf{Case (ii)}} Consider the case when $B\geq\delta$. It is easy to observe that the
condition \eqref{eq-Gener:convex-Main:ineq-gamma>0(2)} is true when
\begin{align*}
\left(\dfrac{1}{\mu}+\delta-1-\xi\right)\geq2B,
\end{align*}
which clearly implies when $B\leq\delta$. Hence this case is not valid.
With the availability of the conditions given above we prove the result for
the case $\gamma>0$ $(\mu>0,\nu>0)$ and $\lambda(t)$ defined in
\eqref{eq-Generalized-hypergeometric-fn}.
\begin{theorem}\label{Thm-Gener:Convex-Generalized-hypergeometric-fn:gamma>0}
Let $A,B,C>0$, $1/2\leq\mu\leq1\leq\nu$ and
$1-\frac{1}{\delta}\leq\zeta\leq1-\frac{1}{2\delta}$, for
$1\leq\delta\leq2$. Let $\beta<1$ satisfy
\begin{align*}
\dfrac{\beta-\frac{1}{2}}{1-\beta}=-K\int_0^1 t^{B-1}(1-t)^{C-A-B}\omega(1-t)
q_{\mu,\nu}^\delta(t) dt,
\end{align*}
where $q_{\mu,\nu}^\delta(t)$ is defined by the differential
equation \eqref{eq-Gener:Convex-q:gamma>0}, the constant $K$
and the function $\omega(1-t)$ is given in
\eqref{eq-Generalized-hypergeometric-fn}. Then for
$f(z)\in\mathcal{W}_\beta^\delta(\alpha,\gamma)$, the function
\begin{align*}
H_{A,\,B,\,C}^\delta(f)(z)=\left(K\int_0^1 t^{B-1}(1-t)^{C-A-B}\omega(1-t) \left(\frac{f(tz)}{t}\right)^\delta dt\right)^{1/\delta}
\end{align*}
belongs to $\mathcal{C}_\delta(\zeta)$ for the condition
${(2-\delta)}/{\mu}\geq{\delta}/{\nu}$, if
{\small{
\begin{align*}
C\geq A+B+2\quad{\rm and}\quad
B\leq\min\left\{\dfrac{1}{4}\left(\dfrac{1}{\mu}-3+\delta(3-2\zeta)\right)\,,\,
\dfrac{2}{\left(\delta+1/\mu\right)}\left(\dfrac{(2\delta-1)}{\mu}-\delta+1\right)\right\}.
\end{align*}}}
\end{theorem}
\begin{proof}
In order to prove the result, it is enough to get the
inequalities \eqref{eq-Gener:convex-Main:ineq-gamma>0(2)},
\eqref{eq-Gener:convex-Main:ineq-gamma>0(11)},
\eqref{eq-Gener:convex-Main:ineq-gamma>0(12)} and
\eqref{eq-Gener:convex-Main:ineq-gamma>0(13)} by using the
above hypothesis.
The inequalities \eqref{eq-Gener:convex-Main:ineq-gamma>0(2)}
and \eqref{eq-Gener:convex-Main:ineq-gamma>0(11)} are true if
$(C-A-B)\geq1+2\xi$ and $2B\leq(1/\mu+\delta-1-\xi)$, where
$\xi=1-\delta(1-\zeta)$. Since the parameters $(C-A-B)>2$ and
$4B\leq(1/\mu+\delta-1-2\xi)$, directly implies that these two
inequalities hold. Moreover, to show the existence of
inequality \eqref{eq-Gener:convex-Main:ineq-gamma>0(12)} under
the given hypothesis, it is enough to prove
\begin{align*}
(C-A-B)\left(\dfrac{1}{\mu}+\delta-2B-1-\xi\right)\geq \left(\frac{1}{\mu}+\delta-1\right)
\end{align*}
or equivalently,
\begin{align*}
(C-A-B-2)\left(\dfrac{1}{\mu}+\delta-2B-1-\xi\right)+\left(\frac{1}{\mu}+\delta-1-2\xi-4B\right)\geq0,
\end{align*}
that can be shown easily. Finally, to prove inequality
\eqref{eq-Gener:convex-Main:ineq-gamma>0(13)}, it is sufficient
to get
\begin{align*}
2(\delta-1)\left(\frac{1}{\mu}-1\right)+2(1-B)\left(\frac{1}{\mu}+\delta-B-1\right)\geq\left(\frac{1}{\mu}+\delta-1\right).
\end{align*}
By simple computation, the above condition is true if
\begin{align*}
\left(\frac{2\delta}{\mu}-\dfrac{1}{\mu}-\delta+1\right)-2B\left(\delta+\frac{1}{\mu}\right)\geq0,
\end{align*}
which is clearly true. Hence by the given hypothesis and
Theorem \ref{Thm:Main-Gener:Convex-Decreas}, the result
directly follows.
\end{proof}
So far, the case $\gamma>0$ was discussed in detail. Now, to
apply Theorem \ref{Thm-Gener:Convex-Mainresult} for the case
$\gamma=0$ $(\mu=0,\,\nu=\alpha\geq0)$, it is required to show
that the function
\begin{align*}
a(t):=\dfrac{\delta\left(1-\frac{1}{\alpha}\right)t^{{(\delta-1)}/{\alpha}}\Lambda_{\alpha}^\delta(t)+
t^{1-1/\alpha}\lambda(t)}{(\log(1/t))^{1+2\xi}}
\end{align*}
is decreasing on $t\in(0,1)$, where $\xi=1-\delta(1-\zeta)$,
for $1/2\leq\alpha\leq1$, $0\leq\xi\leq1/2$ and $\delta\geq1$.
Now, differentiating $a(t)$ and on using
\eqref{eq-weighted:Lambda} will give
\begin{align*}
a'(t)=\dfrac{p(t)\,t^{\delta/\alpha-1/\alpha-1}}{(\log(1/t))^{2+2\xi}}\,\,b(t),
\end{align*}
where
\begin{align*}
&b(t):=\delta\left(1-\dfrac{1}{\alpha}\right)\Lambda_\alpha^\delta(t)+\left(1-\dfrac{(\delta-1)\log(1/t)}{p(t)}\right)
t^{1-\delta/\alpha}\lambda(t)+\dfrac{\log(1/t)}{p(t)}t^{2-\delta/\alpha}\lambda'(t)\\
{\rm and}&\\
&p(t):=\frac{1}{\alpha}(\delta-1)\log\frac{1}{t}+(1+2\xi).
\end{align*}
When $\delta\geq1$, $\alpha\in[1/2,1]$ and $\xi\in[0,1/2]$, it
can be easily seen that the function $p(t)\geq1$, for
$t\in(0,1)$. Hence, proving $a'(t)\leq0$ is equivalent of
getting $b(t)\leq0$. Assuming $\lambda(1)=0$ will give
$b(1)=0$. Hence, if $b(t)$ is increasing function of
$t\in(0,1)$, then $a'(t)\leq0$ and this completes the proof.
Now
\begin{align*}
b'(t)=\dfrac{t^{-\delta/\alpha}}{p(t)}&\left[(\delta-1)\lambda(t)\left(\left(\dfrac{1}{\alpha}-1\right)
\log\frac{1}{t}-1-2\xi+\dfrac{(1+2\xi)}{p(t)}\right)\right.\\
&\quad\left.+t\lambda'(t)\left(\left(3-\delta-\dfrac{1}{\alpha}\right)\log\frac{1}{t}+1+2\xi-\dfrac{(1+2\xi)}{p(t)}\right)
+\log\dfrac{1}{t}\,\,t^2\lambda''(t)\right].
\end{align*}
Therefore, $b'(t)\geq0$, if
\begin{align}\label{eq-Gener:convex-Main:equiv(Cond3)gamma0}
&(\delta-1)\lambda(t)\left(\left(\dfrac{1}{\alpha}-1\right)
\log\frac{1}{t}-1-2\xi+\dfrac{(1+2\xi)}{p(t)}\right)\nonumber\\
&+t\lambda'(t)\left(\left(3-\delta-\dfrac{1}{\alpha}\right)\log\frac{1}{t}+1+2\xi-\dfrac{(1+2\xi)}{p(t)}\right)
+\log\dfrac{1}{t}\,\,t^2\lambda''(t)\geq0.
\end{align}
Now, using $\lambda(t)$, $\lambda'(t)$ and $\lambda''(t)$ given
in \eqref{eq-Generalized-hypergeometric-fn},
\eqref{eq-Generalized-hypergeometric-fn-lambda'} and
\eqref{eq-Generalized-hypergeometric-fn-lambda''}, respectively
in inequality \eqref{eq-Gener:convex-Main:equiv(Cond3)gamma0},
will give the corresponding condition as
\begin{align}\label{eq-Gener:convex-MainCond-Gener:hyper-gamma0}
t^2(1-t)^2\,\log\dfrac{1}{t}\,\,\omega''(1-t)+t(1-t)\,\,X_3(t)\,\,\omega'(1-t)+X_4(t)\,\omega(1-t)\geq0
\end{align}
where
{\small{
\begin{align*}
X_3(t):=\log\dfrac{1}{t}\left[(1-t)\left(\dfrac{1}{\alpha}+\delta-2B-1\right)+2(C-A-B)t\right]
-(1+2\xi)(1-t)\left[1-\dfrac{1}{p(t)}\right]
\end{align*}}}
and
\begin{align*}
X_4(t):=&\log\dfrac{1}{t}\left[(1-t)^2\left[(\delta-1)\left(\frac{1}{\alpha}-1\right)+(1-B)\left(\frac{1}{\alpha}+\delta-B-1\right)\right]\right.\\
&\left.+(C-A-B)t\times\left[(1-t)\left(\dfrac{1}{\alpha}+\delta-2B-1\right)+(C-A-B-1)t\right]\right]\\
&+(1+2\xi)(1-t)\left(\dfrac{}{}(B-\delta)(1-t)
-(C-A-B)t\right)\left[1-\dfrac{1}{p(t)}\right].
\end{align*}
As, the functions $\omega(1-t)$, $\omega'(1-t)$ and
$\omega''(1-t)$ are non-negative for all values of $t\in(0,1)$,
therefore to prove inequality
\eqref{eq-Gener:convex-MainCond-Gener:hyper-gamma0}, it is
enough to show
\begin{align*}
X_3(t)\geq0\quad{\rm and }\quad X_4(t)\geq0.
\end{align*}
Now, we divide the proof into two cases:
\noindent{\bf{Case (i)}} Let $0<B\leq \delta$. Since the function
$p(t)$ defined before is non-negative, therefore by a
small adjustment, the inequality $X_3(t)\geq0$ is
valid, if
\begin{align*}
\log\dfrac{1}{t}\left[(1-t)\left(\dfrac{1}{\alpha}+\delta-2B-1\right)+2(C-A-B)t\right]
\geq(1+2\xi)(1-t),
\end{align*}
where the parameter $\xi$ is defined above. It is easy to
see that the right side of the above inequality is
positive, hence applying the condition
\eqref{eq-general:cond-log-1-t}, the inequality is true
when
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma0(11)}
(1-t)\left[2\left(\dfrac{1}{\alpha}+\delta-2B-\xi\right)-3\right]+2t\left[\dfrac{}{}2(C-A-B-\xi)-1\right]\geq0.
\end{align}
By the assumptions $B\leq\delta$ and $(C-A-B)>0$, the
condition $X_4\geq0$, holds good if
\begin{align*}
&\log\dfrac{1}{t}\left((1-t)^2\left[(\delta-1)\left(\frac{1}{\alpha}-1\right)+(1-B)\left(\frac{1}{\alpha}+\delta-B-1\right)\right]\right.\\
&\left.+(C-A-B)t\times\left[(1-t)\left(\dfrac{1}{\alpha}+\delta-2B-1\right)+(C-A-B-1)t\right]\right)\\
\geq&(1+2\xi)(1-t)\left(\dfrac{}{}(\delta-B)(1-t)+(C-A-B)t\right).
\end{align*}
For $t\in(0,1)$, the right side term of the above
inequality is positive, hence in view of the condition
\eqref{eq-general:cond-log-1-t}, the above inequality can
be obtained if
{\small{
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma01}
(1-t)^2\left[2\left(\delta-1\right)\left(\dfrac{1}{\alpha}-1\right)+2(1-B)\left(\dfrac{1}{\alpha}+\delta-B-1\right)-(1+2\xi)(\delta-B)\right]\nonumber\\
+t(1-t)\left[(C-A-B)\left(2\left(\dfrac{1}{\alpha}+\delta-2B-1\right)-(1+2\xi)\right)-2(1+2\xi)(\delta-B)\right]\nonumber\\
+2t^2(C-A-B)(C-A-B-2-2\xi)\geq0.
\end{align}}}
Thus, the condition
\eqref{eq-Gener:convex-Main:ineq-gamma01} is true, if the
coefficients of $t^2$, $t(1-t)$, and $(1-t)^2$ are
positive. Now, it remains to prove the following
inequalities:
{\small{
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma0:11}
2\left(\delta-1\right)\left(\dfrac{1}{\alpha}-1\right)+2(1-B)\left(\dfrac{1}{\alpha}+\delta-B-1\right)-(1+2\xi)(\delta-B)\geq0,
\end{align}
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma0:12}
(C-A-B)\left(2\left(\dfrac{1}{\alpha}+\delta-2B-1\right)-(1+2\xi)\right)-2(1+2\xi)(\delta-B)\geq0,
\end{align}
and
\begin{align}\label{eq-Gener:convex-Main:ineq-gamma0:13}
(C-A-B)(C-A-B-2-2\xi)\geq0
\end{align}}}
where $\xi=1-\delta(1-\zeta)$, for
$\left(1-\frac{1}{\delta}\right)\leq\zeta\leq
\left(1-\frac{1}{2\delta}\right)$ and $\delta\geq1$.
\noindent{\bf{Case (ii)}} $B\geq\delta$. It is easy to note that the condition
\eqref{eq-Gener:convex-Main:ineq-gamma0(11)} is true when
\begin{align*}
4B\leq2\left(\dfrac{1}{\alpha}+\delta-\xi\right)-3,
\end{align*}
which clearly means that $B\leq\delta$. Therefore this case is not valid.
Now, for the case $\gamma=0$ $(\mu=0,\nu=\alpha>0)$ and
$\lambda(t)$ defined in
\eqref{eq-Generalized-hypergeometric-fn}, the following result
is stated as under.
\begin{theorem}\label{Thm-Gener:Convex-Generalized-hypergeometric-fn:gamma0}
Let $A,B,C>0$, $1/2\leq\alpha\leq1$ and
$1-\frac{1}{\delta}\leq\zeta\leq1-\frac{1}{2\delta}$, for
$\delta\geq3$. Let $\beta<1$ satisfy
\begin{align*}
\dfrac{\beta-\frac{1}{2}}{1-\beta}=-K\int_0^1 t^{B-1}(1-t)^{C-A-B}\omega(1-t)
q_{0,\alpha}^\delta(t) dt,
\end{align*}
where $q_{0,\alpha}^\delta(t)$ is defined by the differential
equation \eqref{eq-generalized:convex-q:gamma0}, the constant
$K$ and the function $\omega(1-t)$ is given in
\eqref{eq-Generalized-hypergeometric-fn}. Then for
$f(z)\in\mathcal{W}_\beta^\delta(\alpha,0)$, the function
\begin{align*}
H_{A,\,B,\,C}^\delta(f)(z)=\left(K\int_0^1 t^{B-1}(1-t)^{C-A-B}\omega(1-t) \left(\frac{f(tz)}{t}\right)^\delta dt\right)^{1/\delta}
\end{align*}
belongs to $\mathcal{C}_\delta(\zeta)$, if
{\small{
\begin{align*}
C\geq A+B+3\quad{\rm and}\quad
B\leq\min\left\{\dfrac{1}{2}\left(\dfrac{1}{\alpha}+\delta-2\right)\,,\,
\dfrac{\delta\left(\frac{1}{\alpha}-1\right)}{\left(\frac{1}{\alpha}+\delta-1\right)}\,,\,
\dfrac{1}{4}\left(\dfrac{3}{\alpha}+\delta-6\right)\right\}.
\end{align*}}}
\end{theorem}
\begin{proof}
In order to prove the result, it is enough to show the
inequalities \eqref{eq-Gener:convex-Main:ineq-gamma0(11)},
\eqref{eq-Gener:convex-Main:ineq-gamma0:11},
\eqref{eq-Gener:convex-Main:ineq-gamma0:12} and
\eqref{eq-Gener:convex-Main:ineq-gamma0:13} by using the above
hypothesis. The inequality
\eqref{eq-Gener:convex-Main:ineq-gamma0(11)} is valid if
$(C-A-B)\geq1$ and $2B\leq(1/\alpha+\delta-2)$, and
\eqref{eq-Gener:convex-Main:ineq-gamma0:11} is true when
$\delta\left({1}/{\alpha}-1\right)\geq\left({1}/{\alpha}+\delta-1\right)B$.
Since the parameters $(C-A-B)\geq3$ and conditions on $B$
holds, which directly implies that these two inequalities along
with the condition \eqref{eq-Gener:convex-Main:ineq-gamma0:13}
are true.
Further, to prove inequality
\eqref{eq-Gener:convex-Main:ineq-gamma0:12}, it is sufficient
to get the condition
\begin{align*}
(C-A-B)\left(2\left(\dfrac{1}{\alpha}+\delta-2B-1\right)-(1+2\xi)\right)\geq2(1+2\xi)(\delta-B),
\end{align*}
By simple computation, the above expression is holds, if
\begin{align*}
(C-A-B-3)\left(\dfrac{1}{\alpha}+\delta-2B-2\right)+\left(\dfrac{3}{\alpha}+\delta-4B-6\right)\geq0,
\end{align*}
which is clearly true. Hence by the given hypothesis and
Theorem \ref{Thm:Main-Gener:Convex-Decreas} the result directly
follows.
\end{proof}
Let
\begin{align*}
\lambda(t)=\dfrac{\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c-a-b+1)}
t^{b-1}(1-t)^{c-a-b}{\,}_{2}F_1\left(\!\!\!\!
\begin{array}{cll}&\displaystyle c-a,\quad 1-a
\\
&\displaystyle c-a-b+1
\end{array};1-t\right),
\end{align*}
then the integral operator \eqref{eq-weighted-integralOperator}
defined by the above weight function $\lambda(t)$ is the known
as generalized Hohlov operator denoted by
$\mathcal{H}_{a,\,b,\,c}^\delta$. This integral operator was
considered in the work of A. Ebadian \cite{Aghalary} (see also \cite{DeviGenlStar}). When
$\delta=1$, the reduced integral transform was introduced by Y.
C. Kim and F. Ronning \cite{KimRonning} and studied by several
authors later. The operator $\mathcal{H}_{a,\,b,\,c}^\delta$, with
$a=1$ is the generalized Carlson-Shaffer operator
($\mathcal{L}_{b,\,c}^\delta$) \cite{CarlsonShaffer}.
Using the above operators the following results are obtained.
\begin{theorem}\label{Thm-Generalized-Convex-Hohlov}
Let $a,b,c>0$, $\gamma\geq0$ $(\mu\geq,\nu\geq0)$ and
$1-\frac{1}{\delta}\leq\zeta\leq1-\frac{1}{2\delta}$. Let
$\beta\!<\!1$ satisfy
{\small{
\begin{align}\label{eq-Generalized-convex-beta-Hohlov}
\dfrac{\beta}{1\!-\!\beta}=-\dfrac{\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c\!-\!a\!-\!b\!+\!1)}\int_0^1\!\! t^{b-1}(1\!-\!t)^{c-a-b}
{\,}_{2}F_1\left(\!\!\!\!\!\!
\begin{array}{cll}&\displaystyle c\!-\!a,{\,} 1\!-\!a
\\
&\displaystyle c\!-\!a\!-\!b\!+\!1
\end{array};1\!-\!t\right)
q_{\mu,\nu}^\delta(t)dt,
\end{align}}}
where $q_{\mu,\nu}^\delta(t)$ is defined by the differential
equation \eqref{eq-Gener:Convex-q:gamma>0} for $\gamma>0$, and
\eqref{eq-generalized:convex-q:gamma0} for $\gamma=0$. Then for
$f(z)\in\mathcal{W}_\beta^\delta(\alpha,\gamma)$, the function
$\mathcal{H}_{a,\,b,\,c}^\delta(f)(z)$ belongs to the class
$\mathcal{C}_\delta(\zeta)$, whenever
\begin{enumerate}[{\rm(i)}]
\item
\begin{align*}
&b\leq\min\left\{\dfrac{1}{4}\left(\dfrac{1}{\mu}-3+\delta(3-2\zeta)\right)\,,\,
\dfrac{2}{\left(\delta+1/\mu\right)}\left(\dfrac{(2\delta-1)}{\mu}-\delta+1\right)\right\}\quad {\rm and}\\
&c\geq a+b+2\quad{\rm for}\quad \gamma>0\,\, (1/2\leq\mu\leq1\leq\nu)\quad{\rm and}\quad1\leq\delta\leq2,
\end{align*}
\item
\begin{align*}
&b\leq\min\left\{\dfrac{1}{2}\left(\dfrac{1}{\alpha}+\delta-2\right)\,,\,
\dfrac{\delta\left(\frac{1}{\alpha}-1\right)}{\left(\frac{1}{\alpha}+\delta-1\right)}\,,\,
\dfrac{1}{4}\left(\dfrac{3}{\alpha}+\delta-6\right)\right\}\quad{\rm and}\\
&c\geq a+b+3\quad{\rm for}\quad 1/2\leq\alpha\leq1,\,\,\gamma=0\,\,
{\rm and}\,\,\delta\geq3.
\end{align*}
\end{enumerate}
\end{theorem}
\begin{proof}
Choosing
\begin{align*}
K=\dfrac{\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c\!-\!a\!-\!b\!+\!1)}\quad{\rm and}\quad
\omega(1-t)=
{\,}_{2}F_1\left(\!\!\!\!\!
\begin{array}{cll}&\displaystyle c-a,{\,} 1-a
\\
&\displaystyle c-a-b+1
\end{array};1-t\right),
\end{align*}
in Theorem
\ref{Thm-Gener:Convex-Generalized-hypergeometric-fn:gamma>0}
and \ref{Thm-Gener:Convex-Generalized-hypergeometric-fn:gamma0}
for the case $\gamma>0$ and $\gamma=0$, respectively to get the
required result.
\end{proof}
For $a=1$, Theorem \ref{Thm-Generalized-Convex-Hohlov} lead to
the following particular cases which are of independent interest.
\begin{corollary}
Let $b, c>0$, $\gamma\geq0$ $(\mu\geq0, \nu\geq0)$ and
$1-\frac{1}{\delta}\leq\zeta\leq1-\frac{1}{2\delta}$. Let
$\beta<1$ satisfy
\begin{align*}
\dfrac{\beta}{(1-\beta)}=-\dfrac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1
t^{b-1}(1-t)^{c-b-1}q_{\mu,\nu}^\delta(t)dt,
\end{align*}
where $q_{\mu,\nu}^\delta(t)$ is defined by the differential
equation \eqref{eq-Gener:Convex-q:gamma>0} for $\gamma>0$, and
\eqref{eq-generalized:convex-q:gamma0} for $\gamma=0$. Then for
$f(z)\in\mathcal{W}_\beta^\delta(\alpha,\gamma)$, the function
$\mathcal{L}_{b,\,c}^\delta(f)(z)$ belongs to the class
$\mathcal{C}_\delta(\zeta)$, whenever
\begin{enumerate}[{\rm(i)}]
\item
\begin{align*}
&b\leq\min\left\{\dfrac{1}{4}\left(\dfrac{1}{\mu}-3+\delta(3-2\zeta)\right)\,,\,
\dfrac{2}{\left(\delta+1/\mu\right)}\left(\dfrac{(2\delta-1)}{\mu}-\delta+1\right)\right\}\quad{\rm and}\\
&c\geq b+3\quad {\rm for}\,\,\, \gamma>0\, (1/2\leq\mu\leq1\leq\nu)\,\, and\,\, 1\leq\delta\leq2
\end{align*}
\item
\begin{align*}
&b\leq\min\left\{\dfrac{1}{2}\left(\dfrac{1}{\alpha}+\delta-2\right)\,,\,
\dfrac{\delta\left(\frac{1}{\alpha}-1\right)}{\left(\frac{1}{\alpha}+\delta-1\right)}\,,\,
\dfrac{1}{4}\left(\dfrac{3}{\alpha}+\delta-6\right)\right\}\quad{\rm and}\\
&c\geq b+4\quad {\rm for}\,\,\,1/2\leq\alpha\leq1,\,\,\gamma=0\,\, {\rm and}\,\, \delta\geq3
\end{align*}
\end{enumerate}
\end{corollary}
\begin{corollary}
Let $b,c>0$, $\gamma\geq0$ $(\mu\geq0,\nu\geq0)$ and
$1-\frac{1}{\delta}\leq\zeta\leq1-\frac{1}{2\delta}$. Let
$\beta_0<\beta<1$, where
\begin{align*}
\beta_0=1-\dfrac{1}{\left(1-{\,}
{\,}_6F_5
\left(
\begin{array}{cll}&\displaystyle \quad\quad 1,b,(1+\delta),(2-\xi),\dfrac{\delta}{\mu},\dfrac{\delta}{\nu},
\\
&\displaystyle c,\delta,(1-\xi),\left(1+\dfrac{\delta}{\mu}\right),\left(1+\dfrac{\delta}{\nu}\right)
\end{array}{\,};{\,}-1\right)
\right)}.
\end{align*}
Then, for $f\in\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$, the
function
$\mathcal{L}_{b,c}^\delta(f)(z)\in\mathcal{C}_\delta(\zeta)$,
whenever
\begin{enumerate}[{\rm(i)}]
\item
\begin{align*}
&b\leq\min\left\{\dfrac{1}{4}\left(\dfrac{1}{\mu}-3+\delta(3-2\zeta)\right)\,,\,
\dfrac{2}{\left(\delta+1/\mu\right)}\left(\dfrac{(2\delta-1)}{\mu}-\delta+1\right)\right\}\quad{\rm and}\\
&c\geq b+3\quad {\rm for}\,\,\, \gamma>0\, (1/2\leq\mu\leq1\leq\nu)\,\, and\,\, 1\leq\delta\leq2
\end{align*}
\item
\begin{align*}
&b\leq\min\left\{\dfrac{1}{2}\left(\dfrac{1}{\alpha}+\delta-2\right)\,,\,
\dfrac{\delta\left(\frac{1}{\alpha}-1\right)}{\left(\frac{1}{\alpha}+\delta-1\right)}\,,\,
\dfrac{1}{4}\left(\dfrac{3}{\alpha}+\delta-6\right)\right\}\quad{\rm and}\\
&c\geq b+4\quad {\rm for}\,\,\,1/2\leq\alpha\leq1,\,\,\gamma=0\,\, {\rm and}\,\, \delta\geq3
\end{align*}
\end{enumerate}
\end{corollary}
\begin{proof}
Putting $a=1$ in \eqref{eq-Generalized-convex-beta-Hohlov} and
on further using \eqref{eq-gener:convex-q:hyper:series} will
give
{\small{
\begin{align*}
\dfrac{\beta}{1\!-\!\beta}=-\dfrac{\Gamma(c)}{\Gamma{(b)}\Gamma{(c\!-\!b)}}\int_0^1\!\!t^{b-1}(1\!-\!t)^{c-b-1}
{\,}_5F_4\!
\left(\!\!\!\!\!\!\!\!
\begin{array}{cll}&\displaystyle \quad\quad 1,(1+\delta),(2-\xi),\dfrac{\delta}{\mu},\dfrac{\delta}{\nu}
\\
&\displaystyle \delta,(1\!-\!\xi),\left(1\!+\!\dfrac{\delta}{\mu}\right),\left(1\!+\!\dfrac{\delta}{\nu}\right)
\end{array}{\,};{\,}-t\right)dt
\end{align*}}}
or equivalently,
{\small{
\begin{align*}
\dfrac{\beta}{1-\beta}=-\dfrac{\Gamma(c)}{\Gamma{(b)}\Gamma{(c\!-\!b)}}\int_0^1\!\!t^{b-1}(1\!-\!t)^{c-b-1}
\left(\sum_{n=0}^{\infty}\dfrac{(1+\delta)_n(2-\xi)_n\left(\dfrac{\delta}{\mu}\right)_n\left(\dfrac{\delta}{\nu}\right)_n
(-1)^n}{(\delta)_n(1-\xi)_n\left(n+\dfrac{\delta}{\nu}\right)_n\left(n+\dfrac{\delta}{\mu}\right)_n} t^n\right)dt.
\end{align*}}}
Now a simple computation leads to
\begin{align*}
\dfrac{\beta}{1-\beta}
=-{\,}_6F_5\!
\left(\begin{array}{cll}&\displaystyle \quad\quad 1,b,(1+\delta),(2-\xi),\dfrac{\delta}{\mu},\dfrac{\delta}{\nu}
\\
&\displaystyle c,\delta,(1-\xi),\left(1+\dfrac{\delta}{\mu}\right),\left(1+\dfrac{\delta}{\nu}\right)
\end{array}{\,};{\,}-1\right).
\end{align*}
Thus, applying Theorem \ref{Thm-Generalized-Convex-Hohlov} will
give the required result.
\end{proof}
Consider
\begin{align}\label{eq-komatu_operator}
\lambda(t)=\dfrac{(1+k)^p}{\Gamma(p)}t^{k}\left(\log\dfrac{1}{t}\right)^{p-1},
\quad \delta\geq 0\quad k>-1.
\end{align}
Then the integral operator \eqref{eq-weighted-integralOperator}
defined by the above weight function $\lambda(t)$ is the known
as generalized Komatu operator denoted by $(F_{k,\,p}^\delta)$.
This integral operator was considered in the work of A. Ebadian
\cite{Aghalary}. When $\delta=1$, the operator is reduced to
the one introduced by Y. Komatu \cite{komatu}.
Now, we state the following result.
\begin{theorem}
Let $\gamma\geq0$ $(\mu\geq,\nu\geq0)$, $k>-1$, $p\geq1$ and
$1-\frac{1}{\delta}\leq\zeta\leq1-\frac{1}{2\delta}$. Let
$\beta\!<\!1$ satisfy \eqref{Beta-Cond-Generalized:Convex},
where $\lambda(t)$ is given in \eqref{eq-komatu_operator}. Then
for $f(z)\in\mathcal{W}_\beta^\delta(\alpha,\gamma)$, the
function $F_{k,p}^\delta(f)(z)\in\mathcal{C}_\delta(\zeta)$,
whenever
\begin{enumerate}[{\rm(i)}]
\item {\small{
\begin{align*}
&-1<k\leq\min\left\{\dfrac{1}{4}\left(\dfrac{1}{\mu}-3+\delta(3-2\zeta)\right)-1\,,\,
\dfrac{2}{\left(\delta+1/\mu\right)}\left(\dfrac{(2\delta-1)}{\mu}-\delta+1\right)-1\right\}\\
&{\rm and}\quad p\geq 1\,\, {\rm for}\,\, \gamma>0\,(1/2\leq\mu\leq1\leq\nu)\,\, {\rm and}\,\,
1\leq\delta\leq2,
\end{align*}}}
\item {\small{
\begin{align*}
&-1<k\leq\min\left\{\dfrac{1}{2}\left(\dfrac{1}{\alpha}+\delta-4\right)\,,\,
\dfrac{\delta\left(\frac{1}{\alpha}-1\right)}{\left(\frac{1}{\alpha}+\delta-1\right)}-1\,,\,
\dfrac{1}{4}\left(\dfrac{3}{\alpha}+\delta-10\right)\right\}\\
&{\rm and}\quad p\geq 2\,\, {\rm for}\,\, 1/2\leq\alpha\leq1,\,\, \gamma=0\,\,{\rm and}\,\,\delta\geq3.
\end{align*}}}
\end{enumerate}
\end{theorem}
\begin{proof}
Letting $(C-A-B)=p-1$, $B=k+1$ and
$\omega(1-t)=\left(\frac{\log(1/t)}{(1-t)}\right)^{p-1}$.
Therefore $\lambda(t)$ given in
\eqref{eq-Generalized-hypergeometric-fn} can be represented as
\begin{align*}
\lambda(t)=Kt^{k}(1-t)^{p-1}\omega(1-t),\quad{\mbox{where}\quad} K=\dfrac{(1+k)^p}{\Gamma(p)}.
\end{align*}
Now, by the given hypothesis the result directly follows from
Theorem
\ref{Thm-Gener:Convex-Generalized-hypergeometric-fn:gamma>0}
and \ref{Thm-Gener:Convex-Generalized-hypergeometric-fn:gamma0}
for the case $\gamma>0$ and $\gamma=0$, respectively.
\end{proof}
| 200,946
|
\begin{document}
\numberwithin{equation}{section}
\allowdisplaybreaks
\renewcommand{\PaperNumber}{097}
\FirstPageHeading
\renewcommand{\thefootnote}{$\star$}
\ShortArticleName{Dif\/ferential Invariants of Conformal and Projective Surfaces}
\ArticleName{Dif\/ferential Invariants of Conformal \\ and Projective Surfaces\footnote{This paper is a
contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas~P.\ Branson. The full collection is available at
\href{http://www.emis.de/journals/SIGMA/MGC2007.html}{http://www.emis.de/journals/SIGMA/MGC2007.html}}}
\Author{Evelyne HUBERT~$^\dag$ and Peter J. OLVER~$^\ddag$}
\AuthorNameForHeading{E.~Hubert and P.J.~Olver}
\Address{$^\dag$~INRIA, 06902 Sophia Antipolis, France}
\EmailD{\href{mailto:Evelyne.Hubert@inria.fr}{Evelyne.Hubert@inria.fr}}
\URLaddressD{\url{http://www.inria.fr/cafe/Evelyne.Hubert}}
\Address{$^\ddag$~School of Mathematics, University of Minnesota,
Minneapolis 55455, USA}
\EmailD{\href{mailto:olver@math.umn.edu}{olver@math.umn.edu}}
\URLaddressD{\url{http://www.math.umn.edu/~olver}}
\ArticleDates{Received August 15, 2007, in f\/inal form September 24,
2007; Published online October 02, 2007}
\Abstract{We show that, for both the conformal and projective groups, all the
dif\/ferential invariants of a generic surface in three-dimensional
space can be written as combinations of the invariant derivatives of a
single dif\/ferential invariant. The proof is based on the equivariant
method of moving frames.}
\Keywords{conformal dif\/ferential geometry;
projective dif\/ferential geometry;
dif\/ferential invariants;
moving frame;
syzygy;
dif\/ferential algebra}
\Classification{14L30; 70G65;
53A30;
53A20;
53A55;
12H05}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\section{Introduction}
According to Cartan, the local geometry of submanifolds under transformation groups, including equivalence and symmetry properties, are entirely governed by their dif\/ferential invariants. Familiar examples are curvature and torsion of a curve in three-dimensional Euclidean space, and the Gauss and mean curvatures of a surface, \cite{Gug,E,Spivak3}.
In general, given a Lie group $G$ acting on a manifold $M$, we are interested in studying its induced action on submanifolds $S \subset M$ of a prescribed dimension, say $p<m = \dim M$. To this end, we prolong the group action to the submanifold jet bundles ${\rm J}^n = {\rm J}^n(M,p)$ of order $n \geq 0$,~\cite{E}. A \textit{differential invariant} is a (perhaps locally def\/ined) real-valued function $I \colon {\rm J}^n \to {\mathbb R}$ that is invariant under the prolonged group action. Any f\/inite-dimensional Lie group action admits an inf\/inite number of functionally independent dif\/ferential invariants of progressively higher and higher order. Moreover, there always exist $p = \dim S$ linearly independent invariant dif\/ferential operators ${\mathcal D}_1,\dots,{\mathcal D}_p$. For curves, the invariant dif\/ferentiation is with respect to the group-invariant arc length parameter; for Euclidean surfaces, with respect to the diagonalizing Frenet frame, \cite{Gug,KOivb,MB1,MBdicc,MBp}.
The \textit{Fundamental Basis Theorem}, f\/irst formulated by Lie, \cite[p.~760]{LieCG}, states that all the dif\/ferential invariants can be generated from a f\/inite number of low order invariants by repeated invariant dif\/ferentiation. A modern statement and proof of Lie's Theorem can be found, for instance, in \cite{E}.
A basic question, then, is to f\/ind a minimal set of generating dif\/ferential invariants. For curves, where $p=1$, the answer is known:
under mild restrictions on the group action (spe\-cif\/i\-cal\-ly transitivity and no pseudo-stabilization under prolongation), there are exactly \mbox{$m-1$} generating dif\/ferential invariants, and any other dif\/ferential invariant is a function of the generating invariants and their successive derivatives with respect to arc length \cite{E}.
Thus, for instance, the dif\/ferential invariants of a space curve $C \subset {\mathbb R}^3$ under the action of the Euclidean group ${\rm SE}(3)$, are generated by $m-1 = 2$ dif\/ferential invariants, namely its curvature and torsion.
In \cite{Osurf}, it was proved, surprisingly that, for generic surfaces in three-dimensional space under the action of either the Euclidean or equi-af\/f\/ine (volume-preserving af\/f\/ine) groups, a minimal system of generating dif\/ferential invariants consists of a \textit{single} dif\/ferential invariant. In the Euclidean case, the mean curvature serves as a generator of the Euclidean dif\/ferential invariants under invariant dif\/ferentiation. In particular, an explicit, apparently new formula expressing the Gauss curvature as a rational function of derivatives of the mean curvature with respect to the Frenet frame was found. In the equi-af\/f\/ine case, there is a single third order dif\/ferential invariant, known as the Pick invariant, \cite{Simon,Spivak3}, which was shown to generate all the equi-af\/f\/ine dif\/ferential invariants through invariant dif\/ferentiation.
In this paper, we extend this research program to study the dif\/ferential invariants of surfaces in~${\mathbb R}^3$ under the action of the conformal and the projective groups. Tresse classif\/ied the dif\/ferential invariants in both cases in 1894,
\cite{Tressedi}. Subsequent developments in conformal geometry can be found in \cite{AGc,BaEaGr,FefGra, Vessiotc}, as well as the work of Tom Branson and collaborators surveyed in the papers in this special issue, while \cite{AGp,FC,MBO} present results on the projective geometry of submanifolds.
The goal of this note is to prove that, just as in the Euclidean and equi-af\/f\/ine cases, the dif\/ferential invariants of both actions are generated by a single dif\/ferential invariant though invariant dif\/ferentiation with respect to the induced Frenet frame. However, lest one be tempted to na\"ively generalize these results, \cite{Ogdi} gives examples of f\/inite-dimensional Lie groups acting on surfaces in ${\mathbb R}^3$ which require an arbitrarily large number of generating dif\/ferential invariants.
Our two main results are:
\begin{theorem}\label{conformal:th} Every differential invariant of a
generic surface $S \subset {\mathbb R}^3$ under the action of the conformal
group ${\rm SO}(4,1)$ can be written in terms of a single third order
invariant and its invariant derivatives.
\end{theorem}
\begin{theorem}\label{projective:th} Every differential invariant of
a generic surface $S \subset {\mathbb R}^3$ under the action of the
projective group ${\rm PSL}(4)$ can be written in terms of a single
fourth order invariant and its invariant derivatives.
\end{theorem}
The proofs follow the methods developed in \cite{Osurf}.
They are based on \cite{FOmcII}, where moving frames were
introduced as equivariant maps from the manifold to the group.
A recent survey of the many developments and applications
this approach has entailed can be found in \cite{Ochina}.
Further extensions are in \cite{hubert07,hubert07b,hubert08a,hubert08b,Ogdi}.
A moving frame induces an invariantization process that maps dif\/ferential functions and dif\/ferential operators to dif\/ferential invariants and (non-commuting) invariant dif\/ferential operators.
Normalized dif\/ferential invariants are
the invariantizations of the standard jet coordinates
and are shown to generate dif\/ferential invariants at each order:
any dif\/ferential invariant can be written as a function of
the normalized invariants. This rewriting is actually a trivial replacement.
The key to the explicit, f\/inite description of dif\/ferential invariants
of any order lies in the \emph{recurrence formulae} that
explicitly relate the dif\/ferentiated and normalized dif\/ferential
invariants. Those formulae show that any dif\/ferential invariant
can be written in terms of a f\/inite set of normalized dif\/ferential
invariants and their invariant derivatives.
Combined with the replacement rule, the formulae make
the rewriting process ef\/fective.
Remarkably, these fundamental relations can be constructed using only the (prolonged) inf\/initesimal generators of the group action and the moving frame normalization equations. One does \textit{not} need to know the explicit formulas for either the group action, or the moving frame, or even the dif\/ferential invariants and invariant dif\/ferential operators, in order to completely characterize
generating sets of dif\/ferential invariants and their syzygies.
Moreover
the syzygies and recurrence relations are given by rational functions
and are thus amenable to algebraic algorithms and symbolic software
\cite{ncdiffalg,hubert05,aida, hubert07b} that we have used for this paper.
\section[Moving frames and differential invariants]{Moving frames and dif\/ferential invariants}
In this section we review the
construction of dif\/ferential invariants and invariant derivations
proposed in \cite{FOmcII}; see also \cite{hubert07b,hubert08a,Ogdi,Osurf}.
Let $G$ be an $r$-dimensional Lie group that acts (locally) on an $m$-dimensional manifold $M$. We are interested in the
action of $G$ on $p$-dimensional submanifolds $N \subset M$ which, in local coordinates, we identify with the graphs of functions \mbox{$u = f(x)$}. For each positive integer $n$, let $G^{(n)}$ denote the prolonged group action on the associated $n$-th order submanifold jet space ${\rm J}^n = {\rm J}^n(M,p)$, def\/ined as the set of equivalence classes of $p$-dimensional submanifolds of $M$ under the equivalence relation of $n$-th order contact. Local coordinates
on ${\rm J}^n$ are denoted $z^{(n)} = (x,u^{(n)}) = (\ \dots \ x^i \ \dots \ u^\alpha _J \ \dots \ )$, with $u^\alpha _J$ representing the partial derivatives of
the dependent variables $u = (u^1,\dots,u^q)$ with respect to the independent variables $x = (x^1,\dots,x^p)$, where $p+q = m$, \cite{E}.
Assuming that the prolonged action is free\footnote{A theorem of Ovsiannikov, \cite{Ov},
slightly corrected in \cite{Osinmf}, guarantees local freeness of the prolonged action at
suf\/f\/iciently high order, provided $G$ acts locally ef\/fectively on subsets of $M$. This is only a technical
restriction; for example, all analytic actions can be made ef\/fective by dividing by the global isotropy
subgroup. Although all known examples of prolonged ef\/fective group actions are, in fact, free on an open subset
of a suf\/f\/iciently high order jet space, there is, frustratingly, as yet no general proof, nor known
counterexample, to this result.} on an open subset of
${\rm J}^n$, then one can construct a (locally def\/ined) \textit{moving frame}, which, according to
\cite{FOmcII}, is an equivariant map $\rho \colon V^n \to G$ def\/ined on an open subset $V^n \subset {\rm J}^n$. Equivariance can be with respect
to either the right or left multiplication action of $G$ on itself. All classical moving
frames, e.g., those appearing in
\cite{Cartanrm, Greenc, Griffithsmf, Gug, ivey03, Jensen}, can be regarded as left equivariant maps,
but the right equivariant versions may be easier to compute, and will be the version used here. Of course, any right moving frame
can be converted to a left moving frame by composition with the inversion map
$g\mapsto g^{-1}$.
In practice, one constructs a moving frame by the process of normalization, relying on
the choice of a local \textit{cross-section} $K^n \subset {\rm J}^n$ to the prolonged group orbits, meaning a submanifold of the complementary dimension that intersects each orbit transversally. A general cross-section is prescribed implicitly by setting $r = \dim G$ dif\/ferential functions $Z = (Z_1,\dots,Z_r)$ to constants:
\begin{gather}\label{Z:eq}
Z_1(x,u^{(n)}) = c_1, \quad \dots, \quad Z_r(x,u^{(n)}) = c_r.\end{gather}
Usually -- but not always, \cite{Mansfield,Osurf} -- the functions are selected from the jet space coordinates $x^i$, $u^\alpha _J$, resulting in a \textit{coordinate cross-section}.
The corresponding value of
the right moving frame at a jet $z^{(n)} \in {\rm J}^n$ is the unique group element $g = \rho^{(n)}(z^{(n)})\in G$ that maps it to
the cross-section:
\begin{equation}\label{mfn:eq}
\rho^{(n)}(z^{(n)})\cdot z^{(n)} = g^{(n)} \cdot z^{(n)} \in K^n.\end{equation}
The moving frame $\rho^{(n)}$ clearly depends on the
choice of cross-section, which is usually designed so as to simplify
the required computations as much as possible.
Once the cross-section has been f\/ixed, the induced moving frame engenders an invariantization process,
that ef\/fectively maps functions to invariants, dif\/ferential forms to invariant
dif\/ferential forms, and so on, \cite{FOmcII,Ochina}. Geometrically, the \textit{invariantization} of any object
is def\/ined as the unique invariant object that coincides with its progenitor when restricted to the
cross-section.
In the special case of functions,
invariantization is actually entirely
def\/ined by the cross-section, and therefore
doesn't require the action to be (locally) free.
It is a projection from the ring of dif\/ferential
functions to the ring of dif\/ferential invariants,
the latter being isomorphic
to the ring of smooth functions on the cross-section \cite{hubert07b}.
Pragmatically, the invariantization of a dif\/ferential function is constructed by f\/irst writing out how it is transformed by the prolonged group action: $F(z^{(n)}) \mapsto F(g^{(n)} \cdot z^{(n)})$. One then
replaces all the group parameters by their \textit{right} moving frame formulae $g = \rho^{(n)}(z^{(n)})$, resulting in the
dif\/ferential invariant
\begin{gather}\label{iota:eq}
\iota\big[F(z^{(n)})\big] = F\big(\rho^{(n)}(z^{(n)}) \cdot z^{(n)}\big) .
\end{gather}
Dif\/ferential forms and dif\/ferential operators are handled in an analogous fashion~-- see
\cite{FOmcII,KOivb} for complete details.
Alternatively, the algebraic construction for the invariantization
of functions in \cite{hubert07b}
works with the knowledge of the cross-section only,
i.e. without the explicit formulae for the moving frame,
and applies to non-free actions as well.
In particular, the \textit{normalized differential invariants} induced by the moving frame
are obtained by invariantization of the basic jet coordinates:
\begin{gather}\label{HI:eq}
H^i = \iota(x^i) ,\qquad I^\alpha_J= \iota(u^\alpha _J) ,
\end{gather}
which we collectively denote by $(H,I^{(n)}) = (\ \ldots \ H^i \ \ldots \ I^\alpha_J \ \ldots \ )$ for $\#J \leq n$.
In the case of a~coordinate cross-section,
these naturally split into two classes: Those corresponding to the
cross-section functions $Z_\kappa$ are constant,
and known as the \textit{phantom differential
invariants}. The remainder, known as the \textit{basic differential invariants}, form a complete system of functionally independent dif\/ferential invariants.
Once the normalized dif\/ferential invariants are known, the invariantization process \eqref{iota:eq} is implemented by simply
replacing each jet coordinate by the corresponding normalized dif\/ferential
invariant \eqref{HI:eq}, so that
\begin{gather}\label{iotaF:eq}
\iota \big[F(x,u^{(n)})\big] = \iota\big[F(\ \dots \ x^i \ \dots \ u^\alpha _J \ \dots \ )\big]=
F(\ \dots \ H^i \ \dots \ I^\alpha_J \ \dots \ ) = F(H,I^{(n)}).\end{gather}
In particular, a dif\/ferential invariant is not af\/fected by invariantization, leading to the very useful \textit{Replacement Theorem}:
\begin{gather}\label{iotaI:eq}
J(x,u^{(n)}) = J(H,I^{(n)}) \quad \mbox{whenever $J$ is a dif\/ferential invariant.}\end{gather}
This permits one to straightforwardly rewrite any known dif\/ferential invariant
in terms the normalized invariants, and thereby establishes their completeness.
A contact-invariant coframe is obtained by taking the horizontal part
(i.e., deleting any contact forms) of the
invariantization of the basic horizontal one-forms:
\begin{gather}\label{omegai:eq}
\omega ^i \equiv \iota(dx^i) \qquad \mbox{modulo contact forms,}\qquad i=1,\dots,p,
\end{gather}
Invariant dif\/ferential
operators ${\mathcal D}_1,\dots,{\mathcal D}_p$ can then be def\/ined as the associated dual
dif\/ferential operators, def\/ined so that
\[
dF \equiv \sum_{i=1}^p ({\mathcal D}_i F) \, \omega ^i \qquad \mbox{modulo contact forms,}
\]
for any dif\/ferential function $F$.
Details can be found in \cite{FOmcII,KOivb}.
The invariant dif\/ferential operators do not commute in general, but are subject to the commutation formulae
\begin{gather}\label{commutator:eq}
[{\mathcal D}_j,{\mathcal D}_k] = \sum_{i=1}^p Y^i_{jk} \,{\mathcal D}_i ,
\end{gather}
where the coef\/f\/icients $Y^i_{jk} = - Y^i_{kj}$ are certain dif\/ferential invariants
known as the \textit{commutator invariants}.
\section{Recurrence and syzygies}
In general, invariantization and dif\/ferentiation do not commute. By a \textit{recurrence relation}, we mean an equation expressing an invariantly dif\/ferentiated invariant in terms of the basic dif\/ferential invariants. Remarkably, the recurrence relations can be deduced knowing only the (prolonged) inf\/initesimal generators of the group action and the choice of cross-section.
Let ${\bf v}_1,\dots,{\bf v}_r$ be a basis for the inf\/initesimal generators of our transformation group. We prolong each inf\/initesimal generator to
${\rm J}^n$, resulting in the vector f\/ields
\begin{gather}\label{prv:eq}
{\bf v}^{(n)}_\kappa = \sum_{i=1}^p \xi ^i_\kappa (x,u) \frac{\partial}{\partial x^i} +
\sum_{\alpha=1}^q \sum_{j = \# J = 0}^n
\varphi ^\alpha_{J,\kappa } (x,u^{(j)}) \frac{\partial}{\partial u^\alpha _J },
\qquad \kappa=1,\dots,r,
\end{gather}
on ${\rm J}^n$. The coef\/f\/icients
$
\varphi ^\alpha _{J,\kappa } = {\bf v}^{(n)}_\kappa(u^\alpha _J)
$
are given by the prolongation formula, \cite{O,E}:
\begin{gather}\label{phiaJ:eq}
\varphi ^\alpha_{J,\kappa}
= D_J \left(\varphi ^\alpha_\kappa - \sum_{i = 1}^p \xi ^i_\kappa \,u^\alpha_i\right)
+ \sum_{i=1}^p \xi^i_\kappa u^\alpha_{J,i}
,
\end{gather}
where $D_1,\dots,D_p$ are the usual (commuting) total derivative operators, and $D_J = D_{j_1} \cdots D_{j_k}$ the corresponding iterated total derivative.
Given a collection $F = (F_1,\dots,F_k)$ of dif\/ferential functions, let
\begin{gather}\label{vF:eq}
{\bf v}(F) = \big({\bf v}^{(n)}_\kappa(F_j)\big)\end{gather}
denote the $r \times k$ \textit{generalized Lie matrix} obtained by applying the prolonged inf\/initesimal generators to the dif\/ferential functions. In particular, $L^{(n)}(x,u^{(n)}) = {\bf v}(x,u^{(n)})$ is the classical Lie matrix of order $n$ whose entries are the inf\/initesimal generator coef\/f\/icients $\xi ^i_\kappa$, $\varphi ^\alpha_{J,\kappa }$,~\cite{E,Ogdi}. The rank of the classical Lie matrix $L^{(n)}(x,u^{(n)})$ equals the dimension of the prolonged group orbit passing through the point $(x,u^{(n)}) \in {\rm J}^n$. We set
\begin{gather}\label{rn:eq}
r_n = \max \big\{{\rm rank}\, L^{(n)}(x,u^{(n)})\,|\,(x,u^{(n)}) \in {\rm J}^n\big\}
\end{gather}
to be the maximal prolonged orbit dimension. Clearly, $r_0 \leq r_1 \leq r_2 \leq \cdots \leq r = \dim G$, and $r_n = r$ if and only if the action is locally free on an open subset of ${\rm J}^n$. Assuming $G$ acts locally ef\/fectively on subsets, \cite{Osinmf}, this holds for $n$ suf\/f\/iciently large. We def\/ine the \textit{stabilization order} $s$ to be the minimal $n$ such that $r_n = r$. Locally, the number of functionally independent dif\/ferential invariants of order $\leq n$ equals $\dim {\rm J}^n - r_n$.
The fundamental moving frame recurrence formulae were f\/irst established in \cite{FOmcII} and written as follows; see also \cite{Ogdi} for additional details.
\begin{theorem}\label{recurrence:th} The \textit{recurrence formulae} for the normalized differential invariants have the form
\begin{gather}\label{DHI:eq}
{\mathcal D}_iH^j = \delta ^j_i + \sum_{\kappa=1}^r R_i^\kappa \, \iota(\xi ^j_\kappa ) ,\qquad
{\mathcal D}_iI^\alpha _J = I^\alpha _{Ji} + \sum_{\kappa=1}^r R_i^\kappa \, \iota(\varphi ^\alpha _{J,\kappa }),
\end{gather}
where $\delta ^j_i$ is the usual Kronecker delta, and $R_i^\kappa$ are certain differential invariants.
\end{theorem}
The recurrence formulae \eqref{DHI:eq} imply the following commutator syzygies among the normalized dif\/ferential invariants:
\begin{gather}\label{comsyz:eq}
{\mathcal D}_iI^\alpha _{Jj} - {\mathcal D}_jI^\alpha _{Ji} = \sum_{\kappa=1}^r \big[R_i^\kappa \, \iota(\varphi ^\alpha _{Jj,\kappa }) - R_j^\kappa \, \iota(\varphi ^\alpha _{Ji,\kappa })\big],\end{gather}
for all $1 \leq i,j \leq p$ and all multi-indices $J$.
We can show that a subset of these relationships~\eqref{DHI:eq},~\eqref{comsyz:eq} form a
complete set of syzygies, \cite{hubert08a}. By formally manipulating those syzygies, performing dif\/ferential
elimination \cite{diffalg,hubert03d,ncdiffalg,hubert05},
we are able to obtain expressions of
some of the dif\/ferential invariants in terms of the invariant derivatives
of others. This is the strategy for the main results of this paper.
In the case of coordinate cross-section, if we single out the recurrence formulae for the constant \textit{phantom differential invariants} prescribed by the cross-section, the left hand sides are all zero, and hence we obtain a linear algebraic system that can be uniquely solved for the invariants~$R_i^\kappa$.
Substituting the resulting formulae back into the recurrence formulae for the remaining, non-constant basic dif\/ferential invariants leads to a complete system of relations among the normalized dif\/ferential invariants \cite{FOmcII,Ogdi}.
More generally, if we think of the $R_i^\kappa$ as the entries of a $p\times r$ matrix
\begin{gather}\label{mcm:eq}
R = (R_i^\kappa ),
\end{gather}
then they are given explicitly by
\begin{gather}\label{MCm:eq}
R = -\iota\big[D(Z)\,{\bf v}(Z)^{-1}\big],\end{gather}
where $Z = (Z_1,\dots,Z_r)$ are the cross-section functions \eqref{Z:eq}, while
\begin{gather}\label{DZ:eq}
D(Z) = (D_i Z_j)\end{gather}
is the $p \times r$ matrix of their total derivatives.
The recurrence formulae are then covered by the matricial equation \cite{hubert08a}
\begin{gather}\label{urf:eq}
{\mathcal D}(\iota(F)) = \iota(D(F)) + R\>\iota({\bf v}(F)),\end{gather}
for any set of dif\/ferential functions $F = (F_1,\dots,F_k)$.
The left hand side denotes the $p \times k$ matrix
\begin{gather}\label{CDF:eq}
{\mathcal D}(\iota(F)) = ({\mathcal D}_i(\iota(F_j)))
\end{gather}
obtained by invariant dif\/ferentiation.
The invariants $R_i^\kappa$ actually arise in the proof of \eqref{DHI:eq}
as the coef\/f\/icients of the horizontal parts of the pull-back of the Maurer--Cartan forms via the moving frame, \cite{FOmcII}. Explicitly, if $\mu^1,\dots,\mu^r$ are a basis for the Maurer--Cartan forms on $G$ dual to the Lie algebra basis ${\bf v}_1,\dots,{\bf v}_r$, then the horizontal part of their pull-back by the moving frame can be expressed in terms of the contact-invariant coframe \eqref{omegai:eq}:
\begin{gather}\label{mfu:eq}
\gamma^\kappa= \rho^* \mu^\kappa \equiv \sum_{i=1}^p R_i^\kappa \,\omega ^i \qquad \mbox{modulo contact forms.}
\end{gather}
We shall therefore refer to $R_i^\kappa$ as the \textit{Maurer--Cartan invariants}, while $R$ in \eqref{mcm:eq} will be called the \textit{Maurer--Cartan matrix}.
In the case of curves, when $G \subset {\rm GL}(N)$ is a matrix Lie group,
the Maurer--Cartan matrix $R = {\mathcal D} \rho^{(n)}(x,u^{(n)}) \cdot \rho^{(n)}(x,u^{(n)})^{-1}$ can be identif\/ied with the
Frenet--Serret matrix, \cite{Gug, MBp}, with ${\mathcal D}$ the invariant arc-length derivative.
The identif\/ication \eqref{mfu:eq} of the Maurer--Cartan invariants as
the coef\/f\/icients of the (horizontal parts of) the pulled-back
Maurer--Cartan forms can be used to deduce their syzygies,
\cite{hubert08b}. The Maurer--Cartan forms on $G$ satisfy the usual Lie
group structure equations
\begin{gather}\label{mcseq:eq}
d \mu^c = - \sum_{a<b} C_{ab}^c \,\mu^a \wedge \mu^b,\qquad c=1,\dots,r,\end{gather}
where $C_{ab}^c$ are the structure constants of the Lie algebra relative to the basis ${\bf v}_1,\dots,{\bf v}_r$.
It follows that their pull-backs \eqref{mfu:eq} satisfy the same equations:
\begin{gather}\label{hmcseq:eq}
d \gamma^c = - \sum_{a<b} C_{ab}^c\, \gamma^a \wedge \gamma^b,\qquad c=1,\dots,r.
\end{gather}
The purely horizontal components of these identities provide the following syzygies among the Maurer--Cartan invariants, \cite{hubert08b}:
\begin{theorem}\label{mcsyz:th}
The Maurer--Cartan invariants
satisfy the following identities:
\begin{gather}\label{MCsyz:eq}
{\mathcal D}_j(R^i_c) - {\mathcal D}_i(R^j_c)
+ \sum_{1\leq a<b\leq r}
C_{ab}^c\, (R^i_a R^j_b- R^j_aR^i_b)
+ \sum_{k=1}^p Y^i_{jk} \, R^k_c = 0, \end{gather}
for
$1\leq c\leq r$, $1\leq i<j\leq p$, and where
$Y^i_{jk} $ are the commutator invariants \eqref{commutator:eq}.
\end{theorem}
Finally, we note the recurrence formulas for the invariant dif\/ferential forms established in \cite{FOmcII} produce the explicit formulas for the commutator invariants:
\begin{gather}\label{Y:eq}
Y^i_{jk} = \sum_{\kappa=1}^r \sum_{j=1}^p
R_j^\kappa\,\iota (D_j\xi ^i_\kappa) - R^\kappa_k\,\iota (D_k\xi ^i_\kappa).
\end{gather}
\section[Generating differential invariants]{Generating dif\/ferential invariants}
A set of dif\/ferential invariants ${\mathfrak I} = \{I_1,\dots,I_k\}$ is called \textit{generating} if, locally, every dif\/ferential invariant can be expressed as a function of them and their iterated invariant derivatives ${\mathcal D}_J I_\nu$. A key issue is to f\/ind a minimal set of generating invariants, which (except for curves) must be done on a case by case basis. Before investigating the minimality question in the conformal and projective examples, let us state general results characterizing (usually non-minimal) generating systems. These results are all consequences of the recurrence formulae, \eqref{DHI:eq} or \eqref{urf:eq}, that furthermore make the rewriting constructive.
Let
\begin{gather}\label{ninv:eq}
{\mathfrak J}^n = \{ H^1, \ldots, H^p\}
\>\cup\> \{ I^\alpha_J\, |\, \alpha =1,\dots,q, \#J \leq n\} \end{gather}
denote the complete set of normalized dif\/ferential invariants of order $\leq n$. In particular, assuming we choose a cross-section that projects to a cross-section on $M$ (e.g., a minimal order cross-section) then ${\mathfrak J}^0 = \{H^1, \ldots, H^p,I^1,\ldots, I^q\}$ are the ordinary invariants for the action on~$M$. In particular, if, as in the examples treated here, the action is transitive on $M$, the normalized order~$0$ invariants are all constant, and hence are superf\/luous for the following generating systems.
\begin{theorem}\label{cngen:th}
If the moving frame has order $n$, then
the set of normalized differential invariants
${\mathfrak J}^{n+1}$ of order $n+1$ or less forms a
generating set.
\end{theorem}
For cross-section of \textit{minimal order} there is an additional
important set of invariants that is generating. This was proved for coordinate cross-sections in
\cite{Ogdi} and then generalized
in \cite{hubert08a}. For each $k \geq 0$, let $r_k$ denote the maximal orbit dimension of the action of $G^{(k)}$ on ${\rm J}^k$.
\begin{theorem}\label{minorder:th}
Let $Z= (Z_1,\dots,Z_r)$ define
a \textit{minimal order cross-section} in the sense that for each $k = 0, 1, \ldots, s$, where $s$ is the stabilization order,
$Z_k=(Z_1, \ldots, Z_{r_k})$ defines a cross-section
for the action of $G^{(k)}$ on ${\rm J}^k$.
Then ${\mathfrak J}^0 \> \cup \>\mathfrak{Z}$, where
\begin{gather}\label{iZ:eq}
\mathfrak{Z}=
\{\iota (D_i(Z_j)) \,|\, 1\leq i\leq p, \; 1\leq j\leq r \},\end{gather}
form a generating set of differential invariants.
\end{theorem}
Another interesting consequence of Theorem \ref{recurrence:th} observed
in \cite{hubert08b} is
that the Maurer--Cartan invariants
\begin{gather}\label{mcinv:eq}
{\mathfrak R}= \{ R^i_a \,|\, 1\leq i\leq p, \; 1\leq a\leq r\}\end{gather}
also form a generating set when the action is
transitive on $M$. More precisely:
\begin{theorem}\label{mcgen:th} The differential invariants ${\mathfrak J}^0\> \cup \>{\mathfrak R}$ form a generating set.
\end{theorem}
In \cite{Osurf}, the following device for generating the commutator invariants was introduced, and then applied to the dif\/ferential invariants of Euclidean and equi-af\/f\/ine surfaces. We will employ the same trick here.
\begin{theorem}\label{ci:th} Let $I = (I_1,\dots,I_p)$ be a set of differential invariants such that ${\mathcal D}(I)$, cf.~\eqref{CDF:eq}, forms a nonsingular $p\times p$ matrix of differentiated invariants. Then one can express the commutator invariants as rational functions of the invariant derivatives, of order $\leq 2$, of $I_1,\dots,I_p$.
\end{theorem}
\begin{proof}
In view of \eqref{commutator:eq}, we have
\begin{gather}\label{DijIk:eq}
{\mathcal D}_i {\mathcal D}_j I_l - {\mathcal D}_j {\mathcal D}_i I_l = \sum_{k=1}^p Y^i_{jk} \,{\mathcal D}_k I_l.\end{gather}
We regard \eqref{DijIk:eq} as a system of $p$ linear equations for the
commutator invariants $Y^i_{j1}, \ldots, Y^i_{jp}$. Our assumption
implies that the coef\/f\/icient matrix is nonsingular. Solving the linear
system by, say, Cramer's rule, produces the formulae for the $Y^i_{jk}$.
\end{proof}
In particular, if $I$ is any single dif\/ferential invariant with suf\/f\/iciently many nontrivial invariant derivatives,
the dif\/ferential invariants in the proposition can be taken as invariant derivatives of~$I$.
Typically we choose $I$ of order at least $n$, the order of the moving frame, and $p-1$ of its f\/irst order invariant derivatives.
If $I$ is a basic invariant, nonsingularity of the matrix of dif\/ferentiated invariants is then a consequence of the recurrence formulae.
As a result, one is, in fact, able to generate all of the commutator invariants as combinations of derivatives of a \textit{single differential invariant\/}!
\section[Differential invariants of surfaces]{Dif\/ferential invariants of surfaces}
Let us specialize the preceding general constructions to the case of two-dimensional surfaces in three-dimensional space. Let $G$ be a $r$-dimensional Lie group acting transitively and ef\/fectively on $M = {\mathbb R}^3$.
Let ${\rm J}^n = {\rm J}^n({\mathbb R}^3,2)$ denote the $n$-th order surface jet bundle, with the usual induced coordinates $z^{(n)} = (x,y,u,u_x,u_y,u_{xx}, \dots , u_{jk}, \dots )$ for $j+k \leq n$.
Let $n \geq s$, the stabilization order of $G$. Given a cross-section $K^n \subset {\rm J}^n$, let $\rho\: \colon V^n \to G$ be the induced right moving frame def\/ined on a suitable open subset $V^n \subset {\rm J}^n$ containing $K^n $.
Invariantization of the basic jet coordinates results in the \textit{normalized differential invariants}
\begin{gather}\label{ndi:eq}
H_1 = \iota(x),\qquad H_2 = \iota(y),\qquad I_{jk} = \iota(u_{jk}),\qquad j, k \geq 0.
\end{gather}
In view of our transitivity assumption, we will only consider cross-sections that normalize the order $0$ variables, $x = y = u = 0$, and so the order $0$ normalized invariants are trivial: $H_1 = H_2 = I_{00} = 0$.
We use
\begin{gather}\label{In:eq}
I^{(n)} = (0,I_{10},I_{01},I_{20},I_{11}, \dots ,I_{0n}) = \iota(u^{(n)})\end{gather}
to denote all the normalized dif\/ferential invariants, both phantom and basic, of order $\leq n$ obtained by invariantizing the dependent variable $u$ and its derivatives.
In addition, the two invariant dif\/ferential operators
${\mathcal D}_1$, ${\mathcal D}_2$, are obtained
as the total derivations dual
to the contact-invariant coframe determined by the moving frame:
Specializing the general moving frame recurrence formulae in Theorem~\ref{recurrence:th}, we have:
\begin{theorem}\label{rf:th} The \textit{recurrence formulae} for the differentiated invariants are
\begin{gather}\label{rf:eq}
{\mathcal D}_1 I_{jk} = I_{j+1,k} + \sum_{\kappa=1}^r \varphi _\kappa ^{jk}(0,0,I^{(j+k)}) R^\kappa_1,\nonumber\\
{\mathcal D}_2 I_{jk} = I_{j,k+1} + \sum_{\kappa=1}^r \varphi _\kappa ^{jk}(0,0,I^{(j+k)}) R^\kappa_2 ,
\qquad j+k \geq 1,
\end{gather}
where $R^\kappa_i$ are the Maurer--Cartan invariants, which multiply the invariantizations of the coefficients of the prolonged infinitesimal generator
\begin{gather}\label{v:eq}
{\bf v}_{\kappa} = \xi_\kappa (x,y,u) \frac{\partial}{\partial x} + \eta_\kappa(x,y,u) \frac{\partial}{\partial y} + \sum_{0 \leq j+k \leq n} \varphi _\kappa^{jk}(x,y,u^{(j+k)})\frac{\partial}{\partial u_{jk}},\end{gather}
which are given explicitly by the usual prolongation formula \eqref{phiaJ:eq}:
\begin{gather}\label{vnc:eq}
\varphi_\kappa ^{jk} = D_x^jD_y^k (\varphi _\kappa- \xi_\kappa\, u_x - \eta _\kappa\, u_y) + \xi _\kappa\, u_{j+1,k} + \eta _\kappa\, u_{k,j+1}.\end{gather}
\end{theorem}
\section{Surfaces in conformal geometry}
In this section, we focus our attention on the standard action of the conformal group ${\rm SO}(4,1)$
on surfaces in ${\mathbb R}^3$, \cite{AGc}. Note that $\dim \hbox{SO}(4,1) = 10$.
A basis for its inf\/initesimal generators is
\begin{gather*}
\frac{\partial}{\partial x}, \qquad \frac{\partial}{\partial y},\qquad \frac{\partial}{\partial u}, \qquad
x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}, \qquad x\frac{\partial}{\partial u}-u\frac{\partial}{\partial x},\qquad y\frac{\partial}{\partial u}-u\frac{\partial}{\partial y}, \\ x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+ u\frac{\partial}{\partial u},\qquad
(x^2-y^2-u^2) \frac{\partial}{\partial x}+2xy\frac{\partial}{\partial y}+2x u \frac{\partial}{\partial u},\\
2 xy\frac{\partial}{\partial x}+ (y^2-x^2-u^2) \frac{\partial}{\partial y} +2yu \frac{\partial}{\partial u},\qquad
2xu\frac{\partial}{\partial u}+2yu \frac{\partial}{\partial y}+ (u^2-x^2-y^2) \frac{\partial}{\partial u}.
\end{gather*}
The maximal prolonged orbit dimensions \eqref{rn:eq} are $r_0=3$, $r_1=5$, $r_2=8$ and $r_3=10$. The stabilization order is thus $s=3$.
The action is transitive on an open subset of ${\rm J}^2$ and there
are two independent dif\/ferential invariants of order $3$.
Thus, by~Theorem~\ref{cngen:th}, the dif\/ferential invariants of
order $3$ and $4$ form a generating set.
In this section we shall show that, under a certain non-degeneracy condition,
all the dif\/ferential invariants can be written in terms of the
derivatives of a single third order dif\/ferential invariant.
The argument goes in two steps. We f\/irst show that all the
dif\/ferential invariants of fourth order can be written in terms of
the two third order dif\/ferential invariants and their monotone derivatives, i.e.~those
obtained by applying the operators ${\mathcal D}_1^i {\mathcal D}_2^j$. Then, the commutator trick of Theorem~\ref{ci:th} allows us to reduce to a single generator.
We give two computational proofs of the f\/irst step. First
using the properties of normalized invariants,
Theorems~\ref{recurrence:th}~and~\ref{minorder:th},
and a cross-section that corresponds to a hyperbolic quadratic form,
second by using the properties of the Maurer--Cartan invariants,
Theorems~\ref{mcgen:th} and~\ref{mcsyz:th}, along with a cross-section that corresponds to a
degenerate quadratic form.
We have used the symbolic computation
software \textsc{aida} \cite{aida} to compute the
Maurer--Cartan matrix, the commutation rules and the
syzygies, and the software \emph{diffalg} \cite{diffalg,ncdiffalg}
to operate the dif\/ferential elimination.
\subsection{Hyperbolic cross-section}
The cross-section implicitly used in \cite{Tressedi} is:
\begin{gather}\label{hypcs:eq}
x=y=u=u_x=u_y=u_{xx}=u_{yy}=u_{xxy}=u_{xyy}=0,\qquad u_{xy} = 1.
\end{gather}
Thus, there are two basic third order dif\/ferential invariants:
\[
I_{30} = \iota(u_{xxx}),\qquad I_{03} = \iota(u_{yyy}),
\]
and $5$ of order $4$, given by invariantization of the fourth order jet coordinates: $I_{jk} = \iota(u_{jk})$, $j+k=4$. Since \eqref{hypcs:eq} def\/ines a minimal order cross-section, Theorem~\ref{minorder:th} implies that
$\{I_{30}, I_{03},I_{31},I_{22},I_{13}\}$ is a generating set of dif\/ferential invariants.
To prove Theorem~\ref{conformal:th}, we f\/irst show that $I_{31}$, $I_{13}$ and $I_{22}$ can be written in terms of
$\{I_{30},I_{03}\}$ and their monotone derivatives.
Using formula~\eqref{MCm:eq}, the Maurer--Cartan matrix is found to have the form
\[
R =
- \begin{pmatrix}
1&0&0&\phi&0&1&0&\kappa&\sigma&\phi
\\
0&1&0&\psi&1&0&0&\sigma&\tau&-\psi\end{pmatrix}
\]
where
\begin{gather*}
\phi=-\tfrac{1}{4}\,I_{{30}},\qquad
\psi=\tfrac{1}{4}\,I_{{03}},\qquad \tau=1-\tfrac{1}{2}\,I_{{13}}-\tfrac{1}{8}\,{I_{{03}}}^{2},\\
\sigma=\tfrac{1}{8}\,I_{{30}}I_{{03}}-\tfrac{1}{2}\,I_{{22}},\qquad
\kappa=1-\tfrac{1}{2}\,I_{{31}}-\tfrac{1}{8}\,{I_{{30}}}^{2}.
\end{gather*}
The f\/irst two are, in fact, the commutator invariants since, by \eqref{Y:eq}, the invariant derivations~${\mathcal D}_1$ and ${\mathcal D}_2$ satisfy the
commutation rule:
\begin{gather}\label{comm2:eq}
[{\mathcal D}_2,{\mathcal D}_1] = \phi\, {\mathcal D}_1 + \psi\,{\mathcal D}_2 .
\end{gather}
Implementing \eqref{DHI:eq},~\eqref{comsyz:eq},
we deduce the following relationships among
$\{I_{30}, I_{03}, I_{40}, I_{31},I_{22}$, $I_{13},I_{04}\}$:
\begin{alignat*}{3}
& {E_{301}}: \quad && {\mathcal D}_1(I_{30})-3\,{I_{22}}
+\tfrac{3}{4}\,{I_{30}}\,{I_{03}}-{I_{40}},&
\\
& {E_{302}}: && {\mathcal D}_2(I_{30})-3\,{I_{13}}
-\tfrac{3}{4}\,{{I_{03}}}^{2}+6-{I_{31}},&
\\
& {E_{031}}: && {\mathcal D}_1(I_{03})-3\,{I_{31}}
-\tfrac{3}{4}\,{{I_{30}}}^{2}+6-{I_{13}},&
\\
&{E_{032}}: && {\mathcal D}_2(I_{03})-3\,{I_{22}}
+\tfrac{3}{4}\,{I_{30}}\,{I_{03}}-{I_{04}},&
\\
& {S_{14}}: && {\mathcal D}_2(I_{13})-{\mathcal D}_1(I_{04})
+\tfrac{3}{4}\,{I_{03}}\,{I_{22}}
-\tfrac{1}{4}\,{I_{03}}\,{I_{04}}+{I_{30}}\,{I_{13}},&
\\
&{S_{23}}: && {\mathcal D}_2(I_{22})-{\mathcal D}_1(I_{13})
-\tfrac{3}{2}\,{I_{03}}\,({I_{31}}+{I_{13}})
-\tfrac{1}{4}\,{I_{30}}\,({I_{22}}+{I_{04}})
-\tfrac{1}{4}\,{I_{03}}\,({{I_{30}}}^{2}+{{I_{03}}}^{2}-20),&
\\
&{S_{32}}: && {\mathcal D}_2(I_{31})-{\mathcal D}_1(I_{22})
+\tfrac{1}{4}\,{I_{03}}\,( {I_{40}}+{I_{22}})
+\tfrac{3}{2}\,{I_{30}}\,({I_{13}}+{I_{31}})
+\tfrac{1}{4}\,{I_{30}}\,({{I_{03}}}^{2}
+{{I_{30}}}^{2}-20),&
\\
&{S_{41}}: &&{\mathcal D}_2(I_{40})-{\mathcal D}_1(I_{31})
-{I_{03}}\,{I_{31}}
-\tfrac{3}{4}\,{I_{30}}\,{I_{22}}
+\tfrac{1}{4}\,{I_{30}}\,{I_{40}}.
&
\end{alignat*}
Taking the combination
$E_{302}-3\,E_{031}$
and
$E_{031}-3\,E_{302}$
we obtain:
\begin{gather*}
I_{31}
= \tfrac{3}{2}-\tfrac{1}{8}\,{\mathcal D}_2(I_{30})
+\tfrac{3}{8}\,{\mathcal D}_1(I_{03})
+{\tfrac{3}{32}}\,{(I_{03})}^{2}
-{\tfrac{9}{32}}\,{(I_{30})}^{2}
,
\\
I_{13}
=
\tfrac{3}{2}-\tfrac{1}{8}\,{\mathcal D}_1(I_{03})
+\tfrac{3}{8}\,{\mathcal D}_2(I_{30})
-{\tfrac{9}{32}}\,{(I_{03})}^{2}
+{\tfrac{3}{32}}\,{(I_{30})}^{2}
.
\end{gather*}
Taking the combination
\begin{gather*}
128\,{\mathcal D}_2(S_{32})-48\,{\mathcal D}_1(S_{41})-16\,{\mathcal D}_1(S_{23})
-36\,I_{03}S_{41}-12\,I_{03}S_{23}+108\,I_{30}S_{32} +4\,I_{30}S_{14}
\\
\qquad{}
-48\,{\mathcal D}_1{\mathcal D}_2(E_{301})-16\,{\mathcal D}_2^{2}(E_{302}) +48\,{\mathcal D}_2^{2}(E_{031})+16\,{\mathcal D}_1^{2}(E_{031})
\\
\qquad{}+36\,I_{03}{\mathcal D}_1(E_{031})+88\,I_{30}{\mathcal D}_2(E_{031})
-12\,I_{30}{\mathcal D}_1(E_{301})-4\,I_{03}{\mathcal D}_2(E_{301})
\\
\qquad{}
+36\,I_{30}{\mathcal D}_2(E_{302})+\left( 18\,{I_{03}}^{2}
+40\,{I_{30}}^{2}+48\,{\mathcal D}_2(I_{30})+24\,{\mathcal D}_1(I_{03})
\right) E_{031}
\\
\qquad{}
+ \left( 18\,I_{30}I_{03}-12\,{\mathcal D}_1(I_{30})+32\,{\mathcal D}_2(I_{03}) \right) E_{301}
\\
\qquad{}
+ \left( 42\,{I_{30}}^{2}+48\,{\mathcal D}_2(I_{30}) \right) E_{302}
+ \left( 2\,I_{30}I_{03}+4\,{\mathcal D}_1(I_{30}) \right)E_{032}
\end{gather*}
leads to:
\[
I_{22} = 1-\frac{A_{22}}{64\,B_{22}},
\]
where
\begin{gather*}
A_{22} =
64\,{\mathcal D}_2^{3}(I_{30})
-\,48\,{\mathcal D}_1^{2}{\mathcal D}_2(I_{30})-48\,{\mathcal D}_1{\mathcal D}_2^{2}(I_{03})
-64\,{\mathcal D}_1^{3}(I_{03}) \\
\phantom{A_{22} =}{} + \left( 36\,{\mathcal D}_1^{2}(I_{03})
+48\,{\mathcal D}_2^{2}(I_{03})
-52\,{\mathcal D}_1{\mathcal D}_2(I_{30})
\right) I_{03}
\\
\phantom{A_{22} =}{}- \left(
36\,{\mathcal D}_2^{2}(I_{30})+24\,{\mathcal D}_1^{2}(I_{30})
-28\,{\mathcal D}_1{\mathcal D}_2(I_{03}) \right) I_{30}
\\
\phantom{A_{22} =}{} +
36\,{{\mathcal D}_2(I_{03})}^{2}
-24\,{{\mathcal D}_1(I_{30})}^{2}+24\,{{\mathcal D}_1(I_{03})}^{2}-24\,{{\mathcal D}_2(I_{30})}^{2}
-12\,{\mathcal D}_2(I_{30}){\mathcal D}_1(I_{03})
\\
\phantom{A_{22} =}{} +
\left(30\,{\mathcal D}_1(I_{03}) -8\,{\mathcal D}_2(I_{30}) \right) {I_{03}}^{2}
+ \left(52\,{\mathcal D}_2(I_{03}) -42\,{\mathcal D}_1(I_{30}) \right)
I_{30}{ I_{03}}
\\
\phantom{A_{22} =}{} - \left(30\,{\mathcal D}_2(I_{30})+2\,{\mathcal D}_1(I_{03}) \right)
{{ I_{30}}}^{2}
+ 3\,{I_{03}}^{4}-3\,I_{30}^{4}+3\,{I_{03}}^{2}-3\,I_{30}^{2},
\end{gather*}
and
\[
B_{22} = {{\mathcal D}_1(I_{30})-{\mathcal D}_2(I_{03})}.
\]
We conclude that the two third order invariants $I_{30}$ and $I_{03}$ form a generating system. Moreover, since the generating invariants are, up to constant multiple, commutator invariants, we can use the commutator trick of Theorem~\ref{ci:th} to generate them both from any single dif\/ferential invariant. Indeed, when ${\mathcal D}_2\phi\neq 0$
the commutation rule \eqref{comm2:eq} implies
that
\begin{gather}\label{psi:eq}
\psi = \frac{{\mathcal D}_2{\mathcal D}_1\phi
-{\mathcal D}_1{\mathcal D}_2\phi -\phi{\mathcal D}_1\phi}{{\mathcal D}_2\phi}.
\end{gather}
Similarly, when ${\mathcal D}_1\psi\neq 0$ we have
\begin{gather}\label{phi:eq}
\phi = \frac{{\mathcal D}_2{\mathcal D}_1\psi
-{\mathcal D}_1{\mathcal D}_2\psi-\psi{\mathcal D}_2\psi}{{\mathcal D}_1\psi}.
\end{gather}
Therefore, under the assumption that
\begin{gather}\label{p1p20:eq}
({\mathcal D}_1\psi)^2+({\mathcal D}_2\phi)^2\neq 0,
\end{gather}
a single dif\/ferential invariant, of order~3,
generates all the dif\/ferential invariants for surfaces in conformal geometry.
\subsection{Degenerate cross-section}
In our second approach, we choose the ``degenerate'' cross-section
\begin{gather}\label{degcs:eq}
x=y=u=u_x=u_y=u_{xx}=u_{xy} = u_{yy}=u_{xxy}=u_{xyy}=0.
\end{gather}
Implementing \eqref{MCm:eq}, the new Maurer--Cartan matrix is:
\[
R =
-\begin{pmatrix}
1&0&0&0&1&0&-\psi&\sigma&\kappa&0
\\ 0&1&0&0&0&0&\phi&\tau&-\sigma&-\tfrac 12\,\phi\end{pmatrix},
\]
where
\[ \phi=I_{{03}},\qquad \psi=I_{{30}},\qquad\tau=\tfrac{1}{2}\,I_{{13}},
\qquad \kappa=-\tfrac{1}{2}\,I_{{31}},\qquad\sigma=\tfrac{1}{2}\,I_{{22}}.
\]
Again, $\phi$, $\psi$ are the commutator invariants since $
[{\mathcal D}_2,{\mathcal D}_1] = \phi\, {\mathcal D}_1 + \psi\,{\mathcal D}_2 $.
Theorem~\ref{mcgen:th} tells us that the Maurer--Cartan invariants $\{\phi,\psi,\kappa,\tau,\sigma\}$ form a generating set.
We will show that $\{\kappa, \tau, \sigma\}$ can be written in terms of
$\{\phi,\psi\}$ and their derivatives. We write those as $\phi_{{ij}}$
to mean ${\mathcal D}^i_1{\mathcal D}^j_2( \phi)$
and similarly for $\psi$, $\kappa$, $\tau$, $\sigma$.
The non-zero syzygies of Theorem~\ref{mcsyz:th} are:
\begin{alignat*}{3}
& \Delta_{7}:\quad &&
\phi_{{10}}+\psi_{{01}}-2\,\tau+2\,\kappa = 0,&\\
& \Delta_{8}:&&
\sigma_{{01}}-\tau_{{10}}-\tfrac{1}{2}\,\phi -2\,\phi \,\sigma -2\,\psi \,\tau =0,&\\
& \Delta_{9}: && \sigma_{{10}}+\kappa_{{01}}-2\,\phi \,\kappa
+2\,\psi \,\sigma =0,&\\
& \Delta_{10}:& & \tfrac{1}{2}\,\phi_{{10}}-\tau
+\psi \,\phi =0.&
\end{alignat*}
The syzygies $ \Delta_{10}$
and $ { \Delta_{7}}+2\,{\Delta_{10}}$
allow us to rewrite $\tau$ and $\kappa$ in terms of $\phi, \psi$, namely:
\[
\kappa=-\tfrac{1}{2}\,\psi_{{01}}+\psi\phi,
\qquad
\tau=\tfrac{1}{2}\,\phi_{{10}}+\psi\phi,
\]
while the following combination
\[
2\,{\mathcal D}_2({\Delta_{9}})-2\,{\mathcal D}_1({\Delta_{8}})
+4\,\sigma{\Delta_{7}}-6\,\psi{\Delta_{8}}
-6\,\phi{\Delta_{9}}-2\,{\Delta_{10}}
\]
allows to express $\sigma$ in terms of
$\phi$, $\psi$, $\tau$, $\kappa$ and their derivatives:
\[
\sigma =
{\frac {\tau_{{20}}+\kappa_{{02}} = 5\,\psi\tau_{{10}}-5\,\phi\kappa_{{01}}
+2\,\psi_{{10}}\tau-2\,\phi_{{01}}\kappa + 6\,{\phi}^{2}\kappa
+(6\,{\psi}^{2}+1)\,\tau
+\frac 12\,\psi\phi}{4(\kappa-\tau)}}.
\]
Observe that this exhibits a singular behavior at \textit{umbilic points} where $\kappa = \tau $.
Finally, since the generating invariants $\{\phi,\psi\}$
are, up to a constant multiple, commutator invariants, we can generate one from the other by the same formulas \eqref{psi:eq},~\eqref{phi:eq}, under the assumption that
\eqref{p1p20:eq} holds.
\section{Projective surfaces}
The inf\/initesimal generators of the projective action of ${\rm PSL}(4)$ on ${\mathbb R}^3$ are
\begin{gather*} \frac{\partial}{\partial x}, \qquad \frac{\partial}{\partial y},\qquad \frac{\partial}{\partial u},
\qquad
x \frac{\partial}{\partial x}, \qquad y \frac{\partial}{\partial x},\qquad
u \frac{\partial}{\partial x}, \\
x \frac{\partial}{\partial y}, \qquad y \frac{\partial}{\partial y},\qquad u \frac{\partial}{\partial y}, \qquad
x \frac{\partial}{\partial u}, \qquad y \frac{\partial}{\partial u},\qquad u \frac{\partial}{\partial u},\\
x^2 \frac{\partial}{\partial x} + xy \frac{\partial}{\partial y}+xu\frac{\partial}{\partial u}, \qquad
x y \frac{\partial}{\partial x} + y^2 \frac{\partial}{\partial y}+ y u\frac{\partial}{\partial u}, \qquad
x u \frac{\partial}{\partial x} + y u \frac{\partial}{\partial y}+ u^2 \frac{\partial}{\partial u} .
\end{gather*}
The generic prolonged orbit dimensions are $r_0=3$, $r_1=5$, $r_2=8$, $r_3=12$ and $r_4=15 = \dim {\rm PSL}(4)$, and so the stabilization order is $s=4$.
We adopt the same strategy as in previous section to show that all
the dif\/ferential invariants are generated by a single fourth order
dif\/ferential invariants. The computations and formulae are nonetheless
more challenging.
The section implicitly used in \cite{Tressedi} is:
\begin{gather}
x=y=u=u_x=u_y=u_{xx}=u_{yy}=u_{xxy}=u_{xyy}=u_{xxxy}=u_{xxyy}=u_{xyyy}=0,\nonumber\\
u_{xy} = u_{xxx} = u_{yyy} =1.\label{prcs:eq}
\end{gather}
Thus, there are two basic fourth order dif\/ferential invariants:
\[
I_{40} = \iota(u_{xxxx}),\qquad I_{04} = \iota(u_{yyyy}),
\]
and $6$ of order $5$, given by invariantization of the f\/ifth order jet coordinates. Theorem~\ref{minorder:th} implies that the invariants $\{I_{40},I_{04},I_{41},I_{32},I_{23},I_{14}\}$ forms a generating set of dif\/ferential invariants.
The Maurer--Cartan matrix \eqref{MCm:eq} is
\begin{gather*}
R =
- \left(\!\begin{array}{ccccccccccccccc}
1 & 0 & 0 & -2\,\psi & 0 & \kappa
& -\tfrac{1}{2} & -\psi & \tau & 0 & 1 & -3\,\psi
& -\tau & \tfrac{1}{4}-\kappa & \tfrac{1}{2}\,\sigma-\tfrac{3}{8}\,\psi
\\
0 & 1 & 0
& \phi & -\tfrac{1}{2} & \sigma
& 0 & 2\,\phi & \eta
& 1 & 0 & 3\,\phi &
\tfrac{1}{4}-\eta & -\sigma & \tfrac{3}{8}\,\phi+\tfrac{1}{2}\,\tau
\end{array}\!\right)\!,\!
\end{gather*}
where
\begin{gather*}
\phi=-\tfrac{1}{3}\,I_{{04}},\qquad \psi=\tfrac{1}{3}\,I_{{40}},\qquad \eta=-\tfrac{1}{2}\,I_{{14}}-\tfrac{1}{4},
\\
\tau=-\tfrac{1}{2}\,I_{{23}}+\tfrac{1}{4}\,I_{{04}},\qquad
\sigma=-\tfrac{1}{2}\,I_{{32}}+\tfrac{1}{4}\,I_{{40}},\qquad
\kappa=-\tfrac{1}{2}\,I_{{41}}-\tfrac{1}{4} .
\end{gather*}
By Theorem~\ref{mcgen:th} $\{\phi,\psi,\tau,\sigma, \kappa \}$
form a generating set of dif\/ferential invariants.
The invariant derivations satisfy the commutation rule;
\[
[{\mathcal D}_2,{\mathcal D}_1] = \phi\, {\mathcal D}_1 + \psi\,{\mathcal D}_2
\]
and so $\phi$, $\psi $ are the commutator invariants.
The nonzero syzygies of Theorem~\ref{mcsyz:th} of the generating set
$\{\phi,\psi, \eta,\sigma, \tau, \kappa\}$ are given by:
\begin{alignat*}{3}
& \Delta_{4}: \quad &&
\phi_{{10}}+2\,\psi_{{01}}+2\,\eta-\phi\,\psi-\tfrac{1}{2} =0,&
\\
& \Delta_6: &&
\sigma_{{10}}-\kappa_{{01}}-\tfrac{3}{8}\,\phi+3\,\phi\,\kappa+2\,\psi\,\sigma=0,&
\\
& \Delta_{8}: & &
2\,\phi_{{10}}+\psi_{{01}}-2\,\kappa+\phi\,\psi+ \tfrac{1}{2}=0,&
\\
& \Delta_{9}: &&
\eta_{{10}}-\tau_{{01}}-\tfrac{3}{8}\,\psi+2\,\phi\,\tau+3\,\psi\,\eta=0,&
\\
& \Delta_{12}: & &
{\Delta_{4}}+{ \Delta_{8}},
\qquad
\Delta_{13}: \ \
- \Delta_{9},
\qquad
\Delta_{14}: \ \
- \Delta_{6} ,&
\\
& \Delta_{15}: & &
\tfrac{1}{2}\,\tau_{{10}}-\tfrac{1}{2}\,\sigma_{{01}}
+\tfrac{3}{8}\,\phi_{{10}}+\tfrac{3}{8}\,\psi_{{01}}-\tfrac{1}{4}\,\kappa+\tfrac{1}{4}
\,\eta+2\,\phi\,\sigma+2\,\psi\,\tau=0.&
\end{alignat*}
From ${\Delta_{4}}$ and ${ \Delta_{8}}$ we immediately obtain:
\[
\eta=
\tfrac{1}{4}-\tfrac{1}{2}\,\phi_{{10}}
-\psi_{{01}}+\tfrac{1}{2}\,\phi\psi,
\qquad
\kappa=
\tfrac{1}{4}+\phi_{{10}}+\tfrac{1}{2}\,\psi_{{01}}+\tfrac{1}{2}\,\phi\psi.
\]
Let $P_1$, $P_2$, $P_3$ be the dif\/ferential polynomials obtained from
$\Delta_6$, $\Delta_9$, $\Delta_{15}$ after substitution of~$\kappa$ and $\tau$:
\begin{gather*}
P_1=-\tfrac{1}{2}\,\tau_{{10}}+\tfrac{1}{2}\,\sigma_{{01}}-2\,\phi\,\sigma-2\,\tau\,\psi,
\\
P_2=\tfrac{1}{2}\,\phi_{{20}}+\psi_{{11}}-\tfrac{1}{2}\,\phi_{{10}}
\psi-\tfrac{1}{2}\,\phi\psi_{{10}}+\tau_{{01}}-\tfrac{3}{8}\,\psi
-2\,\phi\,\tau+\tfrac{3}{2}\,\psi\,\phi_{{10}}+3\,\psi\,\psi_{{01}}
-\tfrac{3}{2}\,\phi\,{\psi}^{2},
\\
P_3=-\sigma_{{10}}+\phi_{{11}}+\phi\phi_{{10}}
+\tfrac{3}{2}\,\psi\phi_{{01}}+\tfrac{1}{2}\,\psi_{{02}}+\tfrac{1}{2}\,\phi
\psi_{{01}}-\tfrac{3}{8}\,\phi-3\,\phi\,\phi_{{10}}
-\tfrac{3}{2}\,\phi\,\psi_{{01}}\\
\phantom{P_3=}{} -\tfrac{3}{2}\,{\phi}^{2}\psi-2\,\psi\,\sigma.
\end{gather*}
To obtain $\tau$ and $\sigma$ we proceed with a \emph{differential elimination}
\cite{diffalg,hubert03d,ncdiffalg,hubert05} on $\{P_1,P_2,P_3\}$. We use a ranking where
\begin{gather*}
\psi<\phi<\psi_{{01}}<\phi_{{01}}<\psi_{{10}}<\phi_{{10}} <
\psi_{{02}}<\psi_{{11}}<\phi_{{11}}<\phi_{{20}}<\>\cdots\\ \qquad
\cdots\><\tau<\sigma <
\tau_{{01}}<\sigma_{{01}}<\tau_{{10}}<\sigma_{{10}}<\tau_{{02}} <
\sigma_{{02}}<\tau_{{11}}<\sigma_{{11}}<\tau_{{20}}<\sigma_{{20}}<\>\cdots\>.
\end{gather*}
For this ranking, the leaders of $P_1$, $P_2$, $P_3$ are, respectively,
$\tau_{{10}}$, $\tau_{{01}}$, $\sigma_{{10}}$.
We f\/irst form the \emph{$\Delta$-polynomial} (cross-derivative) of
$P_1$ and $P_2$ and \emph{reduce} it with respect to
$\{P_1,P_2,P_3\}$. We obtain a polynomial $P_4$ with leader
$\sigma_{{02}}$. We then take the $\Delta$-polynomial of~$P_3$ and
$P_4$ and reduce it with respect to $\{P_1,P_2,P_3,P_4\}$ to obtain a
dif\/ferential polynomial $P_5$ with leader $\sigma_{{01}}$.
On one hand, if we reduce now $P_4$ by $\{P_1,P_2,P_3,P_5\}$ we obtain
a dif\/ferential polynomial~$P$
with leader $\sigma$.
On the other hand, if we form the $\Delta$-polynomial of~$P_3$ and~$P_5$, reduce it by $\{P_1,P_2,P_3,P_5\}$ we obtain a dif\/ferential
polynomial $Q$ with leader $\sigma$. The polynomial~$P$ and~$Q$ are linear in $\sigma$ and $\tau$ so that we can
solve for those two invariants in terms of $\phi$, $\psi$ and their
derivatives. The explicit formulas are rather long (available from the authors on request), but not particularly enlightening. We conclude that the commutator invariants~$\phi$,~$\psi$ form a~generating set. Finally, we can use either \eqref{psi:eq} or \eqref{phi:eq}, to generate one commutator invariant from the other, and thereby establish Theorem~\ref{projective:th}.
\subsection*{Acknowledgements}
This research was initiated during the f\/irst author's visit to the Institute for Mathematics and its Applications (I.M.A.) at the University of Minnesota during 2007--2008 with additional support from the Fulbright visiting scholar program.
The second author is supported in part by NSF Grant DMS 05--05293.
\pdfbookmark[1]{References}{ref}
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\begin{document}
\maketitle
\begin{abstract}
We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity $\lambda$. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^*$.
The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.
\end{abstract}
{\small \noindent
{\bf Keywords:} invariant measure, piecewise-deterministic Markov process, random dynamical system, jump rate, continuous dependence
}\\
{\bf 2010 AMS Subject Classification:} 60J05, 60J25, 37A30, 37A25\\
\section*{Introduction}
Piecewise-deterministic Markov processes (PDMPs) originate with M.H.A. Davis \cite{davis}. They constitute an important class of Markov processes that is complementary to those defined by stochastic differential equations. PDMPs are encountered as suitable mathematical models for processes in the physical world around us, e.g. in resource allocation and service provisioning (queing, cf. \cite{davis}) or biology: as stochastic models for gene expression \cite{tyran}, cell division \cite{lm}, gene regulation \cite{hhs}, excitable membranes \cite{riedler_ea} or population dynamics \cite{alkurdi}.
Mathematical research on PDMPs has been conducted over the years in various directions. Applications in control and optimization have been just one direction. The fundamentals of existence and uniqueness of invariant probability measures for Markov operators and semigroups associated to PDMPs, as well as their asymptotic properties, have attracted much attention. See e.g. \cite{costa2000,costa}, where the considered underlying state space is locally compact. The theory for the general case of non-locally compact Polish state space is less developed yet. It is considered e.g. in \cite{hhs,riedler_ea,dawid,asia,hw}. Another direction is that of establishing the validity of the Strong Law of Large Numbers (SLLN), the Central Limit Theorem (CLT) and the Law of the Interated Logarithm (LIL) for these non-stationary Markov processes (cf. \cite{hhsw,hhsw2,clt_chw,lil_chw}), which has interest in itself for non-stationary processes in general \cite{klo}.
In this paper, we are concerned with a special case of the PDMP described in \cite{dawid,asia}, whose deterministic motion between jumps depends on a~single continuous semi-flow, and any post-jump location is attained by a continuous transformation of the pre-jump state, randomly selected (with a place-dependent probability) among all possible ones. The jumps in this model occur at random time points according to a homogeneous Poisson process.
The random dynamical system of this type constitutes a mathematical framework for certain particular biological models, such as those for gene expression \cite{tyran} or cell division \cite{lm}.
The aim of the paper is to establish the continuous (in the Fortet-Mourier distance, cf. \cite[Section 8.3]{bogachev}) dependence of the invariant measure on the rate of the Poisson process determining the frequency of jumps. While the SLLN and the CLT provide the theoretical foundation for successful approximation of the invariant measure by means of observing or simulating (many) sample trajectories of the process, this result asserts the stability of this procedure, at least locally in parameter space. It is a~prerequisite for the development of a bifurcation theory. Moreover, even stronger regularity of this dependence on parameter (i.e. differentiability in a suitable norm on the space of measures) would be needed for applications in control theory or for parameter estimation (see e.g. \cite{hille_lyczek}).
The outline of the paper is as follows. In Section \ref{sec:1}, several facts on integrating measure-valued functions and basic definitions from the theory of Markov operators have been compiled. Section \ref{sec:model} deals with the structure and assumptions of the model under study. In Section \ref{sec:properties}, we establish certain auxiliary results on the transition operator of the Markov chain given by the post-jump locations. More specifically, we show that the operator is jointly continuous (in the topology of weak convergence of measures) as a function of measure and the jump-rate parameter, and that
the weak convergence of the distributions of the chain to its unique stationary distribution must be uniform. Section \ref{sec:main} is the essential part of the paper. Here, we establish the announced results on the continuous dependence of the invariant measure on the jump frequency for both, the discrete-time system, constituted by the post jump-locations, and for the PDMP itself.
\section{Prelimenaries}\label{sec:1}
Let $X$ be a closed subset of some separable Banach space $(H,\|\cdot\|)$, endowed with the \hbox{$\sigma$-field} $\mathcal{B}_X$ consisting of its Borel subsets. Further, let $(BM(X),\|\cdot\|_{\infty})$ stand for the Banach space of all bounded Borel-measurable functions $f:X\to\mathbb{R}$ with the supremum norm $\|f\|_{\infty}:=\sup_{x\in X} |f(x)|.$ By $BC(X)$ and $BL(X)$ we shall denote the subspaces of $BM(X)$ consisting of all continuous and all Lipschitz continuous functions, respectively. Let us further introduce
\begin{align*}
\|f\|_{BL}:=\max\left\{\|f\|_{\infty}, |f|_{Lip}\right\}\;\;\;\text{for any}\;\;\;f\in BL(X),
\end{align*}
where
\begin{align*}
|f|_{Lip}:=\sup\left\{\frac{|f(x)-f(y)|}{\|x-y\|}:\;x,y\in X,\,x\neq y \right\}.
\end{align*}
It is well-known (cf. \cite[Proposition 1.6.2]{weaver}) that $\|\cdot\|_{BL}$ defines a norm in $BL(X)$, for which it is a~Banach space.
In what follows, we will write $(\mathcal{M}_{sig}(X), \|\cdot\|_{TV})$ for the Banach space of all finite, countably additive functions (signed measures) on $\mathcal{B}_X$, endowed with the total variation norm $\|\cdot\|_{TV}$, which can be expressed~as
\begin{align*}
\|\mu\|_{TV}:=|\mu|(X)=\sup\left\{\left|\left\langle f,\mu\right\rangle\right| :\; f\in BM(X),\,\|f\|_{\infty}\leq 1\right\}\;\;\;\text{for}\;\;\;\mu\in\mathcal{M}_{sig}(X),
\end{align*}
where
\begin{align*}
\left\langle f,\mu\right\rangle:=\int_X f(x)\mu(dx)
\end{align*}
and $|\mu|$ stands for the absolute variation of $\mu$. The symbols $\mathcal{M}_+(X)$ and $\mathcal{M}_1(X)$ will be used to denote the subsets of $\mathcal{M}_{sig}(X)$, consisting of all non-negative and all probability measures on $\mathcal{B}_X$, respectively. Moreover, we will write $\mathcal{M}_{1,1}(X)$ for the set of all measures $\mu\in\mathcal{M}_1(X)$ with finite first moment, i.e. satisfying $\langle\|\cdot\|,\mu\rangle<\infty$.
Let us now define, for any $\mu\in\mathcal{M}_{sig}(X)$, the linear functional $I_{\mu}:BL(X)\to \mathbb{R}$ given by
\begin{align*}
I_{\mu}(f)=\langle f,\mu\rangle\;\;\;\text{for}\;\;\;f\in BL(X).
\end{align*}
It easy to show that $I_{\mu}\in BL(X)^*$ for every $\mu\in\mathcal{M}_{sig}(X)$, where $BL(X)^*$ stands for the dual space of $(BL(X), \|\cdot\|_{BL})$ with the operator norm $\|\cdot\|_{BL}^*$ given by
\begin{align*}
\|\varphi\|_{BL}^*:=\sup\left\{|\varphi(f)|:\; f\in BL(X),\,\|f\|_{BL}\leq 1\right\}\;\;\;\text{for any}\;\;\; \varphi\in BL(X)^*.
\end{align*}
Moreover, we have $\|I_{\mu}\|_{BL}^*\leq\|\mu\|_{TV}$ for any $\mu\in\mathcal{M}_{sig}(X)$.
Furthermore, it is well known (see \cite[Lemma~6]{dudley_baire}), that the mapping
$$\mathcal{M}_{sig}(X)\ni\mu \mapsto I_{\mu}\in BL(X)^*$$
is injective, and thus the space $(\mathcal{M}_{sig}(X), \|\cdot\|_{TV})$ may be embedded into $(BL(X)^*, \|\cdot\|_{BL}^*)$. This enables us to identify each measure $\mu\in \mathcal{M}_{sig}(X)$ with the functional $I_{\mu}\in BL(X)^*$. Note that $\|\cdot\|_{BL}^*$ induces a norm on $\mathcal{M}_{sig}(X)$, called the Fortet-Mourier (or bounded Lipschitz) norm and denoted by $\|\cdot\|_{FM}$. Consequently, we can write
\begin{align*}
\|\mu\|_{FM}:=\left\|I_{\mu}\right\|_{BL}^*=\sup\{|\langle f,\mu\rangle|:\,f\in BL(X),\,\|f\|_{BL}\leq 1\}\;\;\;\text{for any}\;\;\; \mu\in\mathcal{M}_{sig}(X).
\end{align*}
As we have already seen, generally $\|\mu\|_{FM}=\|I_{\mu}\|_{BL}^* \leq \|\mu\|_{TV}$ for any $\mu\in\mathcal{M}_{sig}(X)$. However, for positive measures the norms coincide, i.e. $\|\mu\|_{FM}=\mu(X)=\|\mu\|_{TV}$ for all $\mu\in\mathcal{M}_+(X)$.
Let us now write $\mathcal{D}(X)$ and $\mathcal{D}_+(X)$ for the linear space and the convex cone, respectively, generated by the set $\{\delta_x:\,x\in X\}\subset BL(X)^*$ of functionals of the form
\begin{align*}
\delta_x(f):=f(x)\;\;\;\text{for any}\;\;\;f\in BL(X),\;x\in X,
\end{align*}
which can be also viewed as Dirac measures.
It is not hard to check that the \hbox{$\|\cdot\|_{BL}^*$-closure} of $\mathcal{D}(X)$ is a~separable Banach subspace of $BL(X)^*$. Moreover, assuming that $X$ is complete, one can show
that $\mathcal{M}_+(X)=\text{cl}\,\mathcal{D}_+(X)$ \hbox{(cf. \cite[Theorems 2.3.8--2.3.19]{worm})}, which in turn implies that $\mathcal{M}_{sig}(X)$ is a~\hbox{$\|\cdot\|_{BL}^*$-dense} subspace of $\text{cl}\,\mathcal{D}(X)$, i.e. \hbox{$\text{cl}\,\mathcal{M}_{sig}(X)=\text{cl}\,\mathcal{D}(X)$}. The key idea underlying the proof of this result is to show that every measure $\mu\in\mathcal{M}_+(X)$ can be represented by the Bochner integral (for details see e.g. \cite{vector_measures}) of the continuous map $X\ni x\mapsto \delta_x\in \text{cl}\,\mathcal{D}(X)$, i.e.
\begin{align*}
\mu=\int_X \delta_x\,\mu(dx)\in \text{cl}\, \mathcal{D}_+(X).
\end{align*}
In particular, it follows that $\left(\text{cl}\,\mathcal{M}_{sig}(X),\|\cdot\|_{BL}^*|_{\text{cl}\,\mathcal{D}(X)}\right)$ is a separable Banach space.
What is more, according to \cite[Theorem 2.3.22]{worm}, the dual space of \hbox{$\text{cl}\,\mathcal{M}_{sig}(X)=\text{cl}\,\mathcal{D}(X)$} with the operator norm
\begin{align*}
\|\kappa\|_{\text{cl}\,\mathcal{D}}^{**}:=\sup\{|\kappa(\varphi)|:\,\varphi \in \text{cl}\,\mathcal{D}(X), \; \|\varphi\|_{BL}^*\leq 1\},\;\;\; \kappa\in[\text{cl}\,\mathcal{D}(X)]^*,
\end{align*}
is isometrically isomorphic with the space $(BL(X),\|\cdot\|_{BL})$, and each functional $\kappa\in [\text{cl}\,\mathcal{D}(X)]^*$ can be represented by some $f\in BL(X)$, in the sense that $\kappa(\varphi)=\varphi(f)$ for $\varphi\in \text{cl}\,\mathcal{D}(X)$. In particular, we then have
$\kappa(\mu)=I_{\mu}(f)=\langle f,\mu\rangle$,
whenever $\mu\in\mathcal{M}_{sig}(X)$ (by identyfing $\mu$ with $I_{\mu}$).
In view of the above observations, the norm $\|\cdot\|_{BL}^*$ is convenient for integrating (in the Bochner sense) measure-valued functions $p: E\to\mathcal{M}_{sig}(X)$, where $E$ is an arbitrary measure space. The Pettis measurability theorem (see e.g. \cite[Chapter II, Theorem 2]{vector_measures}), together with the separability of $\text{cl}\,\mathcal{M}_{sig}(X)$, ensures that $p$ is strongly measurable as a map with values in $\text{cl}\,\mathcal{M}_{sig}(X)$ (i.e.~it is a pointwise a.e. limit of simple functions) if and only if, for any $f\in BL(X)$, the functional \hbox{$E\ni t\mapsto\langle f,p(t)\rangle\in\mathbb{R}$} is measurable. Moreover, we have at our disposal the following result (see \cite[Propositions 3.2.3-3.2.5]{worm} or \cite[Proposition C.2]{hille_evers}), which provides a tractable condition guaranteeing the integrability of $p$ and ensuring that the integral is an element of $\mathcal{M}_{sig}(X)$:
\begin{theorem}\label{thm:b_int}
Let $(E,\Sigma)$ be a measurable space with a $\sigma$-finite measure $\nu$, and let \linebreak\hbox{$p:E \to\mathcal{M}_{sig}(X)$} be a strongly measurable function. Suppose that there exists a real-valued function $g\in\mathcal{L}^1(E,\Sigma,\nu)$ such that
\begin{align*}
\|p(t)\|_{TV}\leq g(t)\;\;\;\text{for a.e.}\;\;\; t\in E.
\end{align*}
Then then the following conditions holds:
\begin{itemize}
\item[(i)] The function $p$ is Bochner $\nu$-integrable as a map acting from $(E,\Sigma)$ to \linebreak$\left(\text{cl}\,\mathcal{M}_{sig}(X),\|\cdot\|_{BL}^*|_{\text{cl}\,\mathcal{D}(X)}\right)$. Moreover, we have
$$\left\|\int_E p(t)\,\nu(dt)\right\|_{TV}\leq \int_E \|p(t)\|_{TV}\,\nu(dt).$$
\item[(ii)] The Bochner integral $\int_E p(t)\,\nu(dt)\in\text{cl}\,\mathcal{M}
_{sig}(X)$ belongs to~$\mathcal{M}
_{sig}(X)$ and satisfies
\begin{align*}
\left(\int_E p(t)\nu(dt)\right)(A)=\int_E p(t)(A)\nu(dt)\;\;\;\text{for any}\;\;\; A\in \mathcal{B}_{X}.
\end{align*}
\end{itemize}
\end{theorem}
Another crucial observation is that the restriction of the weak topology on $\mathcal{M}
_{sig}(X)$, generated by $BC(X)$, to $\mathcal{M}_+(X)$ equals to the topology induced by the norm \linebreak\hbox{$\|\cdot\|_{BL}^*|_{\mathcal{M}_+(X)}=\|\cdot\|_{FM}|_{\mathcal{M}_+(X)}$} \hbox{(cf. \cite[Theorem 18]{dudley_baire} or \cite[Theorem 8.3.2]{bogachev})}. In particular, the following holds:
\begin{theorem}
Let $\mu_n,\mu\in\mathcal{M}_+(X)$ for every $n\in\mathbb{N}$. Then $\lim_{n\to\infty} \|\mu_n-\mu\|_{FM}=0$ if and only if $\mu_n\stackrel{w}{\to}\mu$, that is,
$$\lim_{n\to\infty}\langle f,\mu_n\rangle=\left\langle f,\mu\right\rangle\;\;\; \text{for any}\;\;\; f\in BC(X).$$
\end{theorem}
Let us now recall several basic definitions concerning Markov chains. First of all, a~function \hbox{$P:X\times\mathcal{B}_X\to[0,1]$} is called a~stochastic kernel if, for any fixed $A\in \mathcal{B}_X$, \hbox{$x\mapsto P(x,A)$} is a Borel-measurable map on $X$, and, for any fixed $x\in X$, $A\mapsto P(x,A)$ is a~probability Borel measure on $\mathcal{B}_X$. We can consider two operators corresponding to a stochastic kernel $P$, namely
\begin{equation} \label{def:markov}
\mu P(A)=\int_{X} P(x,A)\,\mu(dx)\;\;\;\text{for}\;\;\; \mu\in\mathcal{M}_{sig}(X),\;A\in \mathcal{B}_X
\end{equation}
and
\begin{equation}\label{def:dual}
Pf(x)=\int_{X} f(y)\,P(x,dy)\;\;\;\text{for}\;\;\;f\in BM(X),\; x\in X.
\end{equation}
The operator $(\cdot)P:\mathcal{M}_{sig}(X) \to \mathcal{M}_{sig}(X)$, given by \eqref{def:markov}, is called a~regular Markov operator. It is easy to check that
\begin{equation*}
\langle f, \mu P\rangle=\langle Pf,\mu\rangle\;\;\;\mbox{for any}\;\;\; f\in BM(X),\;\mu\in\mathcal{M}_{sig}(X),
\end{equation*}
and, therefore, $P(\cdot):BM(X)\to BM(X)$, defined by \eqref{def:dual}, is said to be the dual operator of $(\cdot)P$.
A regular Markov operator $(\cdot)P$ is said to be Feller if its dual operator $P(\cdot)$ preserves continuity, that is, $Pf\in BC(X)$ for every $f\in BC(X)$. A measure $\mu^*\in\mathcal{M}_+(X)$ is called an invariant measure for~$(\cdot)P$ whenever $\mu^*P=\mu^*$.
We will say that the operator $(\cdot)P$ is exponentially ergodic in the Fortet-Mourier distance if there exists a unique measure $\mu^*\in\mathcal{M}_1(X)$ invariant of $(\cdot)P$, for which there is $q\in [0,1)$ such that, for any $\mu\in\mathcal{M}_{1,1}(X)$ and some constant $C(\mu)\in\mathbb{R}_+$, we have
$$\left\|\mu P^n-\mu^*\right\|_{FM}\leq C(\mu)q^n\;\;\;\text{for any}\;\;\; n\in\mathbb{N}.$$
The measure $\mu^*$ is then usually called exponentially attracting.
A regular Markov semigroup $({P}(t))_{t\in\mathbb{R}_+}$ is a family of regular Markov operators \linebreak\hbox{$(\cdot){P}(t): \mathcal{M}_{sig}(X) \to \mathcal{M}_{sig}(X)$}, $t\in\mathbb{R}_+:=[0,\infty)$, which form a semigroup (under composition) with the identity transformation $(\cdot){P}(0)$ as the unity element.
Provided that $(\cdot){P}(t)$ is a Markov-Feller operator for every $t\in\mathbb{R}_+$, the semigroup $({P}(t))_{t\in\mathbb{R}_+}$ is said to be Markov-Feller, too. If, for some $\mu^*\in\mathcal{M}_{fin}(X)$, $\mu^*{P}(t)=\mu^*$ for every $t\in\mathbb{R}_+$, then we call $\mu^*$ an invariant measure of $({P}(t))_{t\in\mathbb{R}_+}$.
\section{Description of the model}\label{sec:model}
Recall that $X$ is a closed subset of some separable Banach space $(H,\|\cdot\|)$, and let $(\Theta,\mathcal{B}_{\Theta},\vartheta)$ be a~topological measure space with a $\sigma$-finite Borel measure~$\vartheta$. With a slight abuse of notation, we will further write $d\theta$ only, instead of $\vartheta(d\theta)$.
Let us consider a PDMP $({X}(t))_{t\in\mathbb{R}_+}$, evolving on the space $X$ through random jumps occuring at the jump times $\tau_n$, $n\in\mathbb{N}$, of a homogeneous Poisson process with intensity $\lambda>0$. The state right after the jump is attained by a transformation $w_{\theta}:X\to X$, randomly selected from the set $\{w_{\theta}:\theta\in\Theta\}$. The probability of choosing $w_{\theta}$ is determined by a~place-dependant density function $\theta \mapsto p(x,\theta)$, where $x$ describes the state of the process just before the jump. It is required that the maps \hbox{$(x,\theta)\mapsto p(x,\theta)$} and $(x,\theta)\mapsto w_{\theta}(x)$ are continuous. Between the jumps, the process is deterministically driven by a~continuous (with respect to each variable) semi-flow $S:\mathbb{R}_+\times X\to X$. The flow property means, as usual, that
$S(0,x)=x$ and $S(s+t,x)=S(s,S(t,x))$ for any $x\in X$ and any $s,t\in\mathbb{R}_+$.
Let us now move on to the formal description of the model.
For any $\mu\in\mathcal{M}_1(X)$ and any $\lambda>0$ we first define, on some suitable probability space $(\Omega, \mathcal{F}, \mathbb{P}_{\mu})$, a stochastic proces $(X_n)_{n\in\mathbb{N}_0}$ with initial distribution $\mu$, by setting
\begin{align*}
X_{n+1}=w_{\theta_{n+1}}\left(S\left(\Delta\tau_{n+1},X_{n}\right)\right)\;\;\;\text{for}\;\;\; n\in\mathbb{N}_0,
\end{align*}
with $\Delta\tau_{n+1}=\tau_{n+1}-\tau_n$, where $(\tau_n)_{n\in\mathbb{N}_0}$ and $(\theta_n)_{n\in\mathbb{N}}$ are sequences of random variables with values in $\mathbb{R}_+$ and $\Theta$, respectively, defined in such a way that $\tau_0=0$, $\tau_n \to \infty$ $\mathbb{P}_{\mu}$-a.s., as $n\to\infty$, and
\begin{gather*}
\mathbb{P}_{\mu} \left(\Delta \tau_{n+1} \leq t \,|\, W_n\right)=1-e^{-\lambda t}\;\;\;\text{for any}\;\;\;t\in\mathbb{R}_+,\\
\mathbb{P}_{\mu}\left(\theta_{n+1}\in B\,|\,S\left(\Delta\tau_{n+1},X_n\right)=x,\,W_n\right)=\int_Bp(x,\theta)\,d\theta\;\;\;\text{for any}\;\;\;x\in X,\;B\in\mathcal{B}_{\Theta},
\end{gather*}
with $W_0:=X_0$ and $W_n:=(W_0,\tau_1,\ldots,\tau_n,\theta_1,\ldots,\theta_n)$ for $n\in\mathbb{N}$. We also demand that, for any $n\in\mathbb{N}_0$, the variables $\Delta\tau_{n+1}$ and $\theta_{n+1}$ are conditionally independent given $W_n$.
A standard computation shows that, for any $\lambda>0$, $(X_{n})_{n\in\mathbb{N}_0}$ is a time-homogeneous Markov chain with transition law $P_{\lambda}:X\times \mathcal{B}_X\to[0,1]$ given by
\begin{align}\label{def:Pi_lambda}
P_{\lambda}(x,A)=\int_0^{\infty}\lambda e^{-\lambda t}\int_{\Theta}p(S(t,x),\theta)\,\mathbbm{1}_A\left(w_{\theta}(S(t,x))\right)\,d\theta\,dt\;\;\;\text{for}\;\;\;x\in X,\;A\in\mathcal{B}_X,
\end{align}
that is,
\begin{align*}P_{\lambda}(x,A)=P\left(X_{n+1}\in A\, | \, X_n=x\right) \;\;\; \text{for any} \;\;\;x\in X,\;A\in\mathcal{B}_X.
\end{align*}
On the same probability space, we now define a Markov process $({X}(t))_{t\in\mathbb{R}_+}$, as an iterpolation of the chain $(X_n)_{n\in\mathbb{N}_0}$, namely
\begin{align*}
{X}(t)=S\left(t-\tau_n,X_n\right)\;\;\;\text{for}\;\;\;t\in[\tau_n,\tau_{n+1}),\;\;\;n\in\mathbb{N}_0.
\end{align*}
By $( {P}_{\lambda}(t))_{t\in\mathbb{R}_+}$ we shall denote the Markov semigroup associated with the process $\left({X}(t)\right)_{t\in\mathbb{R}_+}$, so that, for any $t\in\mathbb{R}_+$, ${P}_{\lambda}(t)$ is the Markov operator corresponding to the stochastic kernel satisfying
\begin{align}\label{def:P(t)}
{P}_{\lambda}(t)(x,A)= \mathbb{P}_{\mu}\left({X}(s+t)\in A\, | \, {X}(s)=x\right) \;\;\;\text{for any}\;\;\; A\in\mathcal{B}_X,\; x\in X,\; s \in\mathbb{R}_+.
\end{align}
We further assume that there exist a point $\bar{x}\in X$, a Borel measurable function \hbox{$J:X\to[0,\infty)$} and constants \hbox{$\alpha\in\mathbb{R}$, $L,L_w,L_p,\lambda_{\min},\lambda_{\max},\overline{p}>0$}, such that
\begin{align}\label{constants_ass}
LL_w+\frac{\alpha}{\lambda}<1\;\;\;\text{for each}\;\;\;\lambda\in [\lambda_{\min},\lambda_{\max}],
\end{align}
and, for any $x,y\in X$, the following conditions hold:
\begin{gather}
\sup_{x\in X}\int_0^{\infty} e^{-\lambda_{\min} t}\int_{\Theta} p\left(S(t,x),\theta\right)
\left\|w_{\theta} \left(S(t,\bar{x})\right)\right\|\,d\theta\,dt<\infty,
\label{A1}\\
\left\|S(t,x)-S(t,y)\right\|\leq Le^{\alpha t}\|x-y\|\;\;\;\text{for}\;\;\;t\in\mathbb{R}_+,
\label{A2}\\
\left\|S(t,x)-S(s,x)\right\|\leq\left\{
\begin{array}{ll}
(t-s)e^{\alpha s}J(x),&\text{if}\;\;\alpha\leq 0\\
(t-s)e^{\alpha t}J(x),&\text{if}\;\;\alpha >0
\end{array}
\right.\;\;\;\text{for}\;\;\;0\leq s\leq t,
\label{A6}\\
\int_{\Theta} p(x,\theta)\,\left\|w_{\theta}(x)-w_{\theta}(y)\right\|\,d\theta\leq L_w\left\|x-y\right\|,
\label{A3}\\
\int_{\Theta} |p(x,\theta)-p(y,\theta)|\,d\theta\leq L_{p} \left\|x-y\right\|,
\label{A4}\\
\begin{split}
&\int_{\Theta(x,y)}\min\{p(x,\theta),p(y,\theta)\}\,d\theta\geq \overline{p},
\;\;\;\text{where}\\
&\Theta(x,y):=\{\theta\in\Theta:\,
\left\|w_{\theta}(x)-w_{\theta}(y)\right\|\leq L_w \left\|x-y\right\|\}.
\end{split}\label{A5}
\end{gather}
Note that, if $(H,\left\langle\cdot|\cdot\right\rangle)$ is a Hilbert space and $A:X\to H$ is an $\alpha$-dissipative operator with $\alpha\leq 0$, i.e.
\begin{align*}
\left\langle Ax-Ay|x-y\right\rangle\leq \alpha\left\|x-y\right\|^2\;\;\;\mbox{for any}\;\;\;x,y\in X,
\end{align*}
which additionally satisfies the so-called range condition, that is, for some $T>0$,
\begin{align*}
X\subset \text{Range}\left(\operatorname{id}_X-tA\right)\;\;\;\text{for}\;\;\; t\in(0,T),
\end{align*}
then, for any $x\in X$, the Cauchy problem of the form
\begin{align*}
\left\{\begin{array}{l}
y'(t)=A(y(t))\\
y(0)=x
\end{array}\right.
\end{align*}
has a unique solution $t\mapsto S(t,x)$ such that the semi-flow $S$ enjoys conditions \eqref{A2}, with $L=1$, and \eqref{A6} (cf. \cite[Theorem 5.3, Corollary 5.4]{cl} and \cite[Section 3]{dawid}).
Note that, upon assuming \eqref{constants_ass}, we have $\lambda>\max\{0,\alpha\}$ for any $\lambda\in[\lambda_{\min},\lambda_{\max}]$. Let us further write shortly
\begin{align}\label{def:bar_alpha}
\bar{\alpha}:=\max\{0,\alpha\}.
\end{align}
\section{Some proerties of the operator $P_{\lambda}$}\label{sec:properties}
Consider the abstract model given in Section \ref{sec:model}. In order to simplify notation, for any $t\in\mathbb{R}_+$, let us introduce the function $\Pi_{(t)}:X\times \mathcal{B}_X\to[0,1]$ given by
\begin{align}\label{def:P_phi_t}
\Pi_{(t)}(x,A):=\int_{\Theta}p\left(S(t,x),\theta\right)\,\mathbbm{1}_A\left(w_{\theta}\left(S(t,x)\right)\right)\,d\theta
\;\;\;\text{for}\;\;\;x\in X,\;A\in\mathcal{B}_X.
\end{align}
Note that $\Pi_{(t)}$ is a stochastic kernel, and that the corresponding Markov operator is Feller, due to the continuity of $p$, $S$ and $w_{\theta}$, $\theta\in\Theta$.
Moreover, observe that, for an arbitrary $\lambda>0$, we have
\begin{align}
\begin{split}\label{eq:P_lambda_P_phi}
\mu P_{\lambda}(A)&=\int_X \int_0^{\infty} \lambda e^{-\lambda t} \Pi_{(t)}(x,A)\,dt\,\mu(dx)=\int_0^{\infty}\lambda e^{-\lambda t}\int_X \Pi_{(t)}(x,A) \,\mu(dx)\,dt\\
&=\int_0^{\infty} \lambda e^{-\lambda t} \mu\Pi_{(t)}(A)\,dt\;\;\;\text{for any}\;\;\; \mu\in\mathcal{M}_{sig}(X),\;A\in \mathcal{B}_X.
\end{split}
\end{align}
\begin{lemma}\label{lem:Bochner}
Suppose that conditions \eqref{A6}-\eqref{A4} hold with constants satisfying \eqref{constants_ass}. Then, for any $\lambda>0$ and any $\mu\in\mathcal{M}_{sig}(X)$ satisfying $\langle J,|\mu|\rangle<\infty$, where $J$ is given in \eqref{A6}, the map $t\mapsto e^{-\lambda t}\mu \Pi_{(t)}$ is Bochner integrable on $\mathbb{R}_+$, and we have
\begin{align*}
\mu P_{\lambda}=\int_0^{\infty}\lambda e^{-\lambda t} \mu \Pi_{(t)}\,dt.
\end{align*}
\end{lemma}
\begin{proof}
Let $\mu\in\mathcal{M}_{sig}(X)$ and $\lambda>0$. Note that condition \eqref{A6} implies that
\begin{align*}
\left\|S(t,x)-S(s,x)\right\|\leq J(x)e^{\bar{\alpha}(t+s)}|t-s|\;\;\;\text{for any}\;\;\; s,t\in\mathbb{R}_+,\;x\in X,
\end{align*}
where $\bar{\alpha}$ is given by \eqref{def:bar_alpha}.
Hence, applying \eqref{A3} and \eqref{A4}, we see that, for every \hbox{$f\in BL(X)$},
\begin{align*}
\left|\left\langle f,\mu\Pi_{(t)}\right\rangle-\left\langle f,\mu\Pi_{(s)}\right\rangle\right|
=&\left|\left\langle \Pi_{(t)}f-\Pi_{(s)}f,\mu\right\rangle\right|\\
\leq &\int_X \int_{\Theta}p(S(t,x),\theta)\,\left|f(w_{\theta}(S(t,x)))-f(w_{\theta}(S(s,x)))\right|\,d\theta\,|\mu|(dx)\\
&+\int_X\int_{\Theta}\left|p(S(t,x),\theta)-p(S(s,x),\theta)\right|\,\left|f(w_{\theta}(S(s,x)))\right|\,d\theta\,|\mu|(dx)\\
\leq &\left(|f|_{Lip}L_w+\|f\|_{\infty}L_p\right)\int_X \left\|S(t,x)-S(s,x)\right\|\,|\mu|(dx)\\
\leq &\|f\|_{BL}\left(L_w+L_p\right)\left\langle J,|\mu|\right\rangle e^{\bar{\alpha}(t+s)}|t-s|\;\;\;\text{for}\;\;\;s,t\in\mathbb{R}_+.
\end{align*}
This shows that the map $t\mapsto \left\langle f,e^{-\lambda t}\mu \Pi_{(t)}\right\rangle$ is continuous for any $f\in BL(X)$, and thus it is Borel measurable. Consequently, it now follows from the Pettis theorem (cf. \cite{vector_measures}) that the map $t\mapsto e^{-\lambda t}\mu\Pi_{(t)}$ is strongly measurable. Furthermore, we have
\begin{align*}
\left\|e^{-\lambda t}\mu\Pi_{(t)}\right\|_{TV}
\leq
\left\|\mu\right\|_{TV}e^{-\lambda t}\;\;\;\text{for any}\;\;\; t\in\mathbb{R}_+,
\end{align*}
which, due to Theorem \ref{thm:b_int}, yields that $t\mapsto e^{-\lambda t}\mu\Pi_{(t)}\in\text{cl}\,\mathcal{M}_{sig}(X)$ is Bochner integrable (with respect to the Lebesgue measure) on $\mathbb{R}_+$, and that the integral is a measure in $\mathcal{M}_{sig}(X)$, which satisfies
\begin{align*}
\left(\int_0^{\infty} \lambda e^{-\lambda t} \mu\Pi_{(t)}\,dt\right)(A)=\int_0^{\infty} \lambda e^{-\lambda t} \mu\Pi_{(t)}(A)\,dt\;\;\;\text{for any}\;\;\; A\in \mathcal{B}_X.
\end{align*}
The assertion of the lemma now follows from \eqref{eq:P_lambda_P_phi}.
\end{proof}
\begin{lemma}\label{lem:equicnt}
Let $f\in BL(X)$. Upon assuming \eqref{A2}, \eqref{A3} and \eqref{A4} with constants satisfying \eqref{constants_ass}, we have
\begin{align*}
\left\|\mu\Pi_{(t)}\right\|_{FM}\leq \left(1+\left(L_w+L_p\right)Le^{\alpha t}\right)\left\|\mu\right\|_{FM}\;\;\;\text{for any}\;\;\;\mu\in\mathcal{M}_{sig}(X),\;t\in\mathbb{R}_+.
\end{align*}
\end{lemma}
\begin{proof}
Let $f\in BL(X)$ be such that $\|f\|_{BL}\leq 1$.
Obviously,
$\|\Pi_{(t)}f\|_{\infty}\leq 1$
for every $t\in\mathbb{R}_+$.
Moreover, from conditions \eqref{A2}, \eqref{A3}, \eqref{A4} it follows that $\Pi_{(t)}f\in BL(X)$, and
\begin{align*}
|\Pi_{(t)}f|_{Lip}\leq (L_w+L_p)L e^{\alpha t}\;\;\;\text{for any}\;\;\;t\in\mathbb{R}_+,
\end{align*}
since
\begin{align*}
\begin{aligned}
&\left|\Pi_{(t)}f(x)-\Pi_{(t)}f(y)\right|\\
&\qquad=\left|\int_{\Theta}p\left(S(t,x),\theta\right)f\left(w_{\theta}\left(S(t,x)\right)\right)d\theta-\int_{\Theta}p\left(S(t,y),\theta\right)f\left(w_{\theta}\left(S(t,y)\right)\right)d\theta\right|\\
&\qquad\leq
\left(L_w+L_p\right)\|S(t,x)-S(t,y)\|\\
&\qquad\leq \left(L_w+L_p\right)Le^{\alpha t}\|x-y\|\;\;\;\text{for all}\;\;\;x,y\in X,\;t\in\mathbb{R}_+.
\end{aligned}
\end{align*}
Therefore, for any $\mu\in\mathcal{M}_{sig}(X)$ and any $t\in\mathbb{R}_+$, we obtain
\begin{align*}
\begin{aligned}
\left|\left\langle f,\mu\Pi_{(t)}\right\rangle\right|
=\left|\left\langle \Pi_{(t)}f,\mu\right\rangle\right|
\leq \left\|\Pi_{(t)}f\right\|_{BL}\left\|\mu\right\|_{FM},
\end{aligned}
\end{align*}
which gives the desired conclusion.
\end{proof}
\begin{lemma}\label{lem:mvt}
For any $\lambda_1,\lambda_2>0$, we have
\begin{align*}
\int_0^{\infty}\left|\lambda_1 e^{-\lambda_1 t}-\lambda_2 e^{-\lambda_2 t}\right|dt
\leq \left|\lambda_1-\lambda_2\right|\left(\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\right).
\end{align*}
\end{lemma}
\begin{proof}
Without loss of generality, we may assume that $\lambda_1<\lambda_2$.
Since $1-e^{-x}\leq x$ for every $x\in\mathbb{R}$, we obtain
\begin{align*}
\int_0^{\infty}\left|\lambda_1 e^{-\lambda_1 t}-\lambda_2 e^{-\lambda_2 t}\right|dt
&\leq \lambda_1\int_0^{\infty}\left|e^{-\lambda_1 t}-e^{-\lambda_2 t}\right|dt+\left(\lambda_2-\lambda_1\right)\int_0^{\infty}e^{-\lambda_2 t}dt\\
&\leq \lambda_1\int_0^{\infty}e^{-\lambda_1t}\left(1-e^{-(\lambda_2-\lambda_1)t}\right)dt+\frac{\lambda_2-\lambda_1}{\lambda_2}\\
&\leq \lambda_1\left(\lambda_2-\lambda_1\right)\int_0^{\infty}e^{-\lambda_1 t}t\,dt+\frac{\left(\lambda_2-\lambda_1\right)}{\lambda_2}\\
&= \left|\lambda_1-\lambda_2\right|\left(\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\right),
\end{align*}
which completes the proof.
\end{proof}
\begin{lemma}\label{lem:jointly_cnt}
Let $\mathcal{M}_{sig}(X)$ be endowed with the topology induced by the norm $\|\cdot\|_{FM}$, and suppose that conditions \eqref{A2}-\eqref{A4} hold with constants satisfying \eqref{constants_ass}. Then, the map
\hbox{$(\bar{\alpha},\infty)\times \mathcal{M}_{sig}(X)\ni(\lambda,\mu)\mapsto \mu P_{\lambda}\in\mathcal{M}_{sig}(X),$}
where $\bar{\alpha}$ is given by \eqref{def:bar_alpha},
is jointly continuous.
\end{lemma}
\begin{proof}
Let $\lambda_1,\lambda_2>\bar{\alpha}$ and $\mu_1,\mu_2\in\mathcal{M}_{sig}(X)$. Note that, due to Lemma \ref{lem:Bochner}, we have
\begin{align*}
&\left\|\mu_1 P_{\lambda_1}-\mu_2 P_{\lambda_2}\right\|_{FM}
=\left\|\int_0^{\infty}\left(\lambda_1 e^{-\lambda_1 t}\mu_1\Pi_{(t)}-\lambda_2 e^{-\lambda_2 t}\mu_2\Pi_{(t)}\right)dt\right\|_{FM}\\
&\qquad\qquad\leq \left\|\mu_1\right\|_{TV}\int_0^{\infty}\left|\lambda_1 e^{-\lambda_1 t}-\lambda_2 e^{-\lambda_2 t}\right|\,dt+\int_0^{\infty}\lambda_2 e^{-\lambda_2 t}\left\|\mu_1\Pi_{(t)}-\mu_2\Pi_{(t)}\right\|_{FM}dt,
\end{align*}
where the inequality follows from statement (i) of Theorem \ref{thm:b_int} and the fact that \linebreak\hbox{$\|\mu_1\Pi_{(t)}\|_{FM}\leq\|\mu_1\|_{TV}$}.
Further, applying Lemmas \ref{lem:equicnt} and \ref{lem:mvt}, we obtain
\begin{align*}
\begin{aligned}
\left\|\mu_1 P_{\lambda_1}-\mu_2 P_{\lambda_2}\right\|_{FM}
\leq\,\left\|\mu_1\right\|_{TV}\left|\lambda_1-\lambda_2\right|\left(\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\right)
+\left\|\mu_1-\mu_2\right\|_{FM}
\left(1+\frac{\left(L_w+L_p\right)L\lambda_2}{\lambda_2-\alpha}\right).
\end{aligned}
\end{align*}
We now see that $\|\mu_1 P_{\lambda_1} - \mu_2 P_{\lambda_2} \|_{FM} \to 0$, as $|\lambda_1-\lambda_2|\to 0$ and $\|\mu_1-\mu_2\|_{FM}\to 0$, which completes the proof.
\end{proof}
Suppose that \eqref{A1}, \eqref{A2} and \eqref{A3}-\eqref{A5} hold with constants satisfying \eqref{constants_ass}.
Then, according to \hbox{\cite[Theorem 4.1]{dawid}} (or \hbox{\cite[Theorem 4.1]{asia}}), for any $\lambda\in[\lambda_{\min},\lambda_{\max}]$,
there exists a~unique invariant measure $\mu_{\lambda}^*\in\mathcal{M}_1(X)$ for $P_{\lambda}$ such that
\begin{align}\label{eq:conv_lambda}
\left\|\mu P^n_{\lambda}-\mu_{\lambda}^*\right\|_{FM}\leq C_{\lambda,\mu} \,q_{\lambda}^n\;\;\;\text{for any}\;\;\;\mu\in\mathcal{M}_{1,1}(X),
\end{align}
where $q_{\lambda}\in(0,1)$ and $C_{\lambda,\mu}\in\mathbb{R}_+$ are some constants, depending on the parameter $\lambda$ and the initial measure~$\mu$.
\begin{lemma}\label{lem:uniform_conv}
Suppose that conditions \eqref{A1}, \eqref{A2} and \eqref{A3}-\eqref{A5} hold with constants satisfying \eqref{constants_ass}, and, for any $\lambda \in [\lambda_{min}, \lambda_{max}]$, let $\mu_{\lambda}^*$ stand for the unique invariant probability measure measure of $P_{\lambda}$. Then, the convergence $\|\mu P^n_{\lambda} - \mu_{\lambda}^*\|_{FM}\to 0$ (as $n\to \infty$) is uniform with respect to $\lambda$, whenever \hbox{$\mu\in\mathcal{M}_{1,1}(X)$}.
\end{lemma}
\begin{proof}
In view of \cite[Theorem 4.1]{dawid},
it is sufficient to prove that the convergence is uniform with respect to $\lambda$.
Let us consider the case where $\alpha\leq 0$.
Choose an arbitrary $\lambda\in[\lambda_{\min},\lambda_{\max}]$, and note that, substituting $t=\lambda_{\max}\lambda^{-1} u$, we obtain
\begin{align*}
\mu P_{\lambda}(A)&=\int_X\int_0^{\infty}\lambda e^{-\lambda t}\int_{\Theta}p\left(S(t,x),\theta\right)\,\mathbbm{1}_A\left(w_{\theta}\left(S(t,x)\right)\right)\,d\theta\,dt\,\mu(dx)\\
&=\int_X\int_0^{\infty}\lambda_{\max} e^{-\lambda_{\max} u}\int_{\Theta}p\left(S_{\lambda}(u,x),\theta\right)\,\mathbbm{1}_A\left(w_{\theta}\left(S_{\lambda}(u,x)\right)\right)\,d\theta\,du\,\mu(dx)
\end{align*}
for any $\mu\in\mathcal{M}_1(X)$, $A\in\mathcal{B}_X$, where
\begin{align*}
S_{\lambda}(u,x)=S\left(\frac{\lambda_{\max}}{\lambda} u,x\right)\;\;\;\text{for}\;\;\;u\in\mathbb{R}_+,\;x\in X.
\end{align*}
Moreover, the semi-flow $S_{\lambda}$ enjoys condition \eqref{A2}, since, for any $t\in\mathbb{R}_+$ and any $x,y\in X$, we have
\begin{align*}
\left\|S_{\lambda}(t,x)-S_{\lambda}(t,y)\right\|\leq Le^{\alpha\lambda_{\max}\lambda^{-1} t}\left\|x-y\right\|\leq L e^{\alpha t}\left\|x-y\right\|.
\end{align*}
Hence, we can write $P_{\lambda}=\widetilde{P}_{\lambda_{\max}}$, where $\widetilde{P}_{\lambda_{\max}}$ stands for the Markov operator corresponding to the instance of our system with the jump intensity $\lambda_{\max}$ and the flow $S_{\lambda}$ in place of $S$. Taking into account the above observation, it is evident that such a modified system still satisfies conditions \eqref{constants_ass}- \eqref{A2} and \eqref{A3}-\eqref{A5}. Consequently, keeping in mind \eqref{eq:conv_lambda}, we can conclude that there exist $q_{\lambda_{\max}}\in(0,1)$ and $C_{\lambda_{\max},\mu} \in\mathbb{R}_+$ such that, for any $\lambda\in[\lambda_{\min},\lambda_{\max}]$, we have
\begin{align*}
\left\|\mu P^n_{\lambda}-\mu_{\lambda}^*\right\|_{FM}
=\left\|\mu\widetilde{P}_{\lambda_{\max}}^n - \mu_{\lambda}^* \right\|_{FM}
\leq C_{\lambda_{\max},\mu} \,q_{\lambda_{\max}}^n\;\;\;\text{for any}\;\;\;\mu\in\mathcal{M}_{1,1}(X).
\end{align*}
In the case where $\alpha>0$, the proof is similar to the one conducted above (except that this time we substitute $t:=\lambda_{\min}\lambda^{-1} u$), so we omit~it.
\end{proof}
\section{Main results}\label{sec:main}
Before we formulate and prove the main theorems of this paper, let us first quote the result provided in \cite[Theorem 7.11]{rudin}.
\begin{lemma}\label{lem:rudin}
Let $(Y,\varrho)$ and $(Z,d)$ be some metric spaces, and let $U$ be an arbitrary subset of $Y$. Suppose that $(f_n)_{n\in\mathbb{N}_0}$ is a sequence of functions, defined on $E$, with values in $Z$, which converges uniformly on $E$ to some function $f:E\to Z$. Further, let $\bar{y}$ be a limit point of $E$, and assume that
\begin{align*}
a_n:=\lim_{y\to \bar{y}}f_n(y)
\end{align*}
exists and is finite for every ${n\in\mathbb{N}_0}$.
Then, $f$ has a finite limit at $\bar{y}$, and the sequence $(a_n)_{n\in\mathbb{N}_0}$ converges to it, that is,
\begin{align*}
\lim_{n\to\infty}\left(\lim_{y\to \bar{y}}f_n(y)\right)=\lim_{y\to \bar{y}}\left(\lim_{n\to\infty}f_n(y)\right).
\end{align*}
\end{lemma}
We are now in a position to state the result concerning the continuous dependence of an invariant measure $\mu^*_{\lambda}$ of $P_{\lambda}$ on the parameter $\lambda$. In the proof we will refer to the lemmas provided in Section \ref{sec:properties}, as well as to Lemma \ref{lem:rudin}.
\begin{theorem}\label{thm:mu}
Suppose that conditions \eqref{A1}-\eqref{A5} hold with constants satisfying \eqref{constants_ass}, and, for any $\lambda \in [\lambda_{min}, \lambda_{max}]$, let $\mu_{\lambda}^*$ stand for the unique invariant probability measure measure of $P_{\lambda}$. Then, for every $\bar{\lambda}\in[\lambda_{\min},\lambda_{\max}]$, we have
$\mu_{\lambda}^* \stackrel{w}{\to} \mu_{\bar{\lambda}}^*$, as $\lambda\to\bar{\lambda}$.
\end{theorem}
\begin{proof}
Let $\bar{\lambda}\in[\lambda_{\min},\lambda_{\max}]$. Due to Lemma \ref{lem:uniform_conv}, we know that, for every $\mu\in\mathcal{M}_1(X)$ and any \hbox{$\lambda\in [\lambda_{\min}, \lambda_{\max}]$}, the sequence $(\mu P_{\lambda}^n)_{n\in\mathbb{N}_0}$ converges weakly to $\mu^*_{\lambda}$, as $n\to\infty$, and the convergence is uniform with respect to $\lambda$.
Further, Lemma \ref{lem:jointly_cnt} yields that $(\bar{\alpha},\infty)\times \mathcal{M}_1(X)\ni(\lambda,\mu)\mapsto \mu P_{\lambda}\in\mathcal{M}_1(X)$ is jointly continuous, provided that $\mathcal{M}_1(X)$ is equipped with the topology induced by the Fortet-Mourier norm. Hence, for any $\mu\in\mathcal{M}_1(X)$ and any $n\in\mathbb{N}_0$, it follows that $\mu P_{\lambda}^n$ converges weakly to $\mu P_{\bar{\lambda}}^n$, as $\lambda\to\bar{\lambda}$.
Finally, according to Lemma \ref{lem:rudin}, we get
\begin{align*}
\lim_{\lambda\to\bar{\lambda}}\left\langle f,\mu_{\lambda}^*\right\rangle
&=\lim_{\lambda\to\bar{\lambda}}\left(\lim_{n\to\infty}\left\langle f,\mu P_{\lambda}^n\right\rangle\right)
=\lim_{n\to\infty}\left(\lim_{\lambda\to\bar{\lambda}}\left\langle f,\mu P_{\lambda}^n\right\rangle\right)
=\lim_{n\to\infty}\left\langle f,\mu P_{\bar{\lambda}}^n\right\rangle
=\left\langle f,\mu_{\bar{\lambda}}^*\right\rangle
\end{align*}
for any $f\in BC(X)$ and any $\mu\in\mathcal{M}_1(X)$, which completes the proof.
\end{proof}
In the final part of the paper we will study the properties of the semigroup of Markov operators $(P_{\lambda}(t))_{t\in\mathbb{R}_+}$, defined by \eqref{def:P(t)}. I order to apply the relevant results of \cite{dawid}, in what follows, we additionally assume that the measure $\vartheta$, given on the set $\Theta$, is finite. Then, according to \cite[Theorem 4.4]{dawid}, for any $\lambda>0$, there is a one-to-one correspondence between invariant measures of the operator $P_\lambda$ and those of the semigroup $(P_{\lambda}(t))_{t\in\mathbb{R}_+}$. Furthermore,
if $\mu^*_{\lambda}\in\mathcal{M}_1(X)$ is a unique invariant measure of $P_{\lambda}$, then $\nu^*_{\lambda}:=\mu_{\lambda}^*G_{\lambda}\in\mathcal{M}_1(X)$, where
\begin{align*}
\mu G_{\lambda}(A)=\int_X\int_0^{\infty}\lambda e^{-\lambda t}\mathbbm{1}_A\left(S(t,x)\right)\,dt\,\mu(dx)\;\;\;\text{for any}\;\;\;\mu\in\mathcal{M}_1(X),\;\;\;A\in\mathcal{B}_X,
\end{align*}
is a unique invariant measure of $(P_{\lambda}(t))_{t\in\mathbb{R}_+}$.
The main result concerning the continuous-time model, which is formulated and proven below, ensures the continuity of the map $\lambda\mapsto\nu^*_{\lambda}$.
\begin{theorem}\label{thm:main}
Suppose that conditions \eqref{A1}-\eqref{A5} hold with constants satisfying \eqref{constants_ass}, and, for any $\lambda \in [\lambda_{min}, \lambda_{max}]$, let $\mu_{\lambda}^*$ stand for the unique invariant probability measure measure of $P_{\lambda}$. Further, assume that $\Theta$ is endowed with a finite Borel measure $\vartheta$. Then, for every $\bar{\lambda}\in[\lambda_{\min},\lambda_{\max}]$, we have
$\nu_{\lambda}^* \stackrel{w}{\to} \nu_{\bar{\lambda}}^*$, as $\lambda\to\bar{\lambda}$.
\end{theorem}
\begin{proof}
Let $\bar{\lambda}\in[\lambda_{\min},\lambda_{\max}]$, and let $f\in BL(X)$ be such that $\|f\|_{BL}\leq 1$. For any \linebreak$\lambda\in[\lambda_{\min},\lambda_{\max}]$, we have
\begin{align*}
\left\langle f,\nu_{\lambda}^*\right\rangle=\left\langle f,\mu_{\lambda}^*G_{\lambda}\right\rangle
=\int_X\int_0^{\infty}\lambda e^{-\lambda t}f\left(S(t,x)\right)\,dt\,\mu_{\lambda}^*(dx),
\end{align*}
whence
\begin{align*}
\begin{aligned}
\left|\left\langle f,\nu_{\lambda}^*-\nu^*_{\bar{\lambda}}\right\rangle\right|
\leq &\int_0^{\infty}\left|\lambda e^{-\lambda t}-\bar{\lambda}e^{-\bar{\lambda}t}\right|\,dt
+
\left|\int_0^{\infty}\bar{\lambda}e^{-\bar{\lambda}t}\left\langle f\circ S(t,\cdot),\mu^*_{\lambda}-\mu^*_{\bar{\lambda}}\right\rangle\,dt\right|.
\end{aligned}
\end{align*}
Note that, due to \eqref{A2}, $f\circ S(t,\cdot)\in BL(X)$ and $\|f\circ S(t,\cdot)\|_{BL}\leq 1+Le^{\alpha t}$, which implies that
\begin{align*}
\begin{aligned}
\left|\int_0^{\infty}\bar{\lambda}e^{-\bar{\lambda}t}\left\langle f\circ S(t,\cdot),\mu^*_{\lambda}-\mu^*_{\bar{\lambda}}\right\rangle\,dt\right|
&\leq \left\|\mu_{\lambda}^*-\mu_{\bar{\lambda}}^*\right\|_{FM}\int_0^{\infty} \bar{\lambda} e^{-\overline{\lambda} t}\left(1+Le^{\alpha t}\right) \,dt \\
&= \left\|\mu_{\lambda}^*-\mu_{\bar{\lambda}}^*\right\|_{FM}\left(
1+\frac{L\bar{\lambda}}{\bar{\lambda}-\alpha}\right).
\end{aligned}
\end{align*}
Combining this and Lemma \ref{lem:mvt}, finally gives
\begin{align*}
\left\|\nu_{\lambda}^*-\nu_{\bar{\lambda}}^*\right\|_{FM}
\leq \left|\lambda-\bar{\lambda}\right|
\left(\frac{1}{\lambda}+\frac{1}{\bar{\lambda}}\right)+c \left\|\mu_{\lambda}^*-\mu_{\bar{\lambda}}^*\right\|_{FM}
\end{align*}
with $c:=1+L\bar{\lambda}(\bar{\lambda}-\alpha)^{-1}$. Hence, referring to the assertion of Theorem \ref{thm:mu}, we obtain
\begin{align*}
\lim_{\lambda\to\bar{\lambda}}\left\|\nu_{\lambda}^*-\nu_{\bar{\lambda}}^*\right\|_{FM}=0,
\end{align*}
and the proof is completed.
\end{proof}
\section{Acknowledgements}
The work of Hanna Wojew\'odka-\'Sci\k{a}\.zko has been supported by the National Science Centre of Poland, grant number 2018/02/X/ST1/01518.
\bibliography{references}
\bibliographystyle{plain}
\end{document}
| 122,935
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| 115,211
|
TITLE: Simple Eigenvalue finding question (by gauss elimination)
QUESTION [3 upvotes]: I saw a method for finding eigenvalues by using Gauss elimination to find an upper triangular matrix, then just taking the diagonal elements as the eigenvalues. It seems to work except for this case:
\begin{bmatrix}
1&1\\
1&2
\end{bmatrix}
if i make it upper triangular by subtracting the first row from the second to get:
\begin{bmatrix}
1&1\\
0&1
\end{bmatrix}
then I would assume the eigenvalues should be 1, but they are not. What am I doing wrong?
REPLY [9 votes]: You are doing nothing wrong. Gaussian elimination does not preserve eigenvalues, and this method simply does not work.
If you are lucky enough to start with an upper triangular matrix, then it is true that the diagonals are the eigenvalues. This is very convenient. Otherwise, you have to go and actually compute them.
For two by two matrices, there is a sort of shortcut (maybe). The eigenvalues completely determine the trace and the determinant. In particular, $\lambda_1 + \lambda_2 = \text{Trace} A$, and $\lambda_1\lambda_2 = \text{Det}A$. So here, we know that $\lambda_1 + \lambda_2 = 3$ and $\lambda_1 \lambda_2 = 1$.
So $\lambda_1 = 1/\lambda_2$, which leads to $\lambda_1^2 - 3\lambda_1 + 1 = 0$. And so $\lambda_1,\lambda_2$ are the two roots of the polynomial $x^2 - 3x + 1$, which are $\frac{3 \pm \sqrt{5}}{2}$. Is that easier? Sometimes.
| 592
|
\begin{document}
\selectlanguage{english}
\maketitle
\begin{abstract}
For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H.~Fan, T.~Jarvis, and Y.~Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory).
We define orbifold versions of the monodromy operator on the quantum (co)homology group, of the Milnor lattice, of the Seifert form and of the intersection form. We also describe some symmetry properties of invariants of invertible polynomials refining the known ones.
\end{abstract}
\section{Introduction} \label{sect:Intro}
In \cite{Ruan_etal}, for a germ $f$ of a quasihomogeneous function with an isolated critical point at
the origin invariant with respect to an appropriate action of a finite abelian group $G$ (an admissible one),
H.~Fan, T.~Jarvis, and Y.~Ruan defined the so-called quantum cohomology group $\calH_{f,G}$. This group
is related to the vanishing cohomology groups of Milnor fibres of restrictions of $f$ to fixed point sets
of elements of $G$. The quantum cohomology group is considered as the main object of the quantum singularity
theory (FJRW-theory). In \cite{Ruan_etal}, the authors study some structures on it which generalize similar
structures in the usual singularity theory.
An important role in singularity theory is played by such concepts as the (integral) Milnor lattice,
the monodromy operator, the Seifert form and the intersection form. Analogues of these concepts have not
yet been considered in the FJRW-theory.
Here we define an orbifold version of the monodromy operator on the quantum (co)homology group
$\calH_{f,G}$ and a lattice $\Lambda_{f,G}$ in $\calH_{f,G}$ which is invariant with respect to
the orbifold monodromy operator and is considered as an orbifold version of the Milnor lattice.
The action of the orbifold monodromy operator on it can be considered as an analogue of the integral
monodromy operator. Moreover, we define orbifold versions of the Seifert form and of the intersection form.
We show that they are related by equations similar to those in the classical case.
To define these concepts we introduce the language of group rings. An appropriate change of the basis in
the group ring allows to give a decomposition of a certain extension of the quantum (co)homology group
$\calH_{f,G}$ into parts isomorphic to (co)homology groups of certain suspensions of the restrictions of
the function $f$ to fixed point sets. This permits to define analogues of the Seifert and intersection form
on this extension. We show that the intersection of this decomposition with the quantum (co)homology group
respects the relations between the monodromy, the Seifert and the intersection form. This defines these
concepts on the quantum (co)homology group.
In the last section, we shall consider some examples. These examples are chosen in the class of so-called invertible polynomials (orbifold
Landau--Ginzburg models in the terminology of \cite{BH1,BH2}). We also consider the Berglund--H\"ubsch--Henningson duality of pairs $(f,G)$ where $f$ is an invertible polynomial and discuss the behaviour of the Milnor lattice under this mirror symmetry.
The authors are grateful to A.~Takahashi for very useful discussions permitting them to understand some
peculiarities in the definitions which they initially missed. We would like to thank the referee for carefully reading our paper and for valuable comments which helped to improve the paper.
\section{Quantum cohomology group} \label{sect:qhg}
Let $f:(\CC^n,0)\to(\CC,0)$ be a germ of a holomorphic function with an isolated critical point at
the origin and let $G$ be a finite abelian group acting faithfully on $(\CC^n,0)$ and preserving $f$.
(Without loss of generality one can assume that the action of $G$ is linear and diagonal.)
An important example is an invertible polynomial $f$ in $n$ variables with a subgroup $G$
of its maximal diagonal symmetry group $G_f$: \cite{BH2}. To a pair $(f,G)$ of this sort,
P.~Berglund, T.~H\"ubsch, and M.~Henningson \cite{BH1,BH2} defined a dual pair $(\widetilde{f}, \widetilde{G})$. (Another description of the dual group was given by Krawitz \cite{Krawitz}.) This construction plays an
important role in mirror symmetry.
For a subgroup $K\subset G$, let $(\CC^n)^K$ be the fixed point set
$\{x\in \CC^n \, \vert \, \forall g\in K: gx=x\}$ and let $n_K$ be the dimension of $(\CC^n)^K$. The restriction $f_{\vert (\CC^n)^K}$ will be denoted by $f^K$.
If $K$ is the cyclic subgroup $\langle g\rangle$ generated by an element $g\in G$,
we shall use the notations $(\CC^n)^g$, $n_g$ and $f^g$.
The Milnor fibre $V_f$ of the germ $f$ is
$f^{-1}(\varepsilon)\cap B_{\delta}^{2n}$ where $0<\vert\varepsilon\vert\ll\delta$ are small enough,
$B_{\delta}^{2n}$ is the ball of radius $\delta$ centred at the origin in $\CC^n$.
The group $G$ acts on the Milnor fibre $V_f$ and thus on its homology and cohomology groups.
\begin{definition} (cf.\ \cite{Ruan_etal})
The {\em quantum cohomology group} of the pair $(f,G)$ is
\begin{equation}\label{quantum_cohomology}
\calH_{f,G}= \bigoplus_{g\in G} \calH_g\,,
\end{equation}
where $\calH_g:=H^{n_g-1}(V_{f^g};\CC)^G= H^{n_g-1}(V_{f^g}/G;\CC)$ is the $G$-invariant part of the
vanishing cohomology group $H^{n_g-1}(V_{f^g};\CC)$ of the Milnor fibre
of the restriction of $f$ to the fixed point set of $g$. If $n_g=1$, this means the cohomology group $\widetilde{H}^0(V_{f^g};\CC)$ reduced modulo a point. (We keep the notations without the tilde not to overload them.) If $n_g=0$, we assume $H^{-1}(V_{f^g};\CC)$ to be one dimensional with the trivial action of $G$. This means that we consider the ``critical point'' of the function of zero variables to be non-degenerate and thus to have Milnor number equal to one and this corresponds to the definition of $\calH_{f,G}$ in the form given in \cite{Ruan_etal}.
\end{definition}
\begin{remarks}
{\bf 1.} One can show that the restriction $f^g$ of $f$ to $(\CC^n)^g$ has an isolated critical
point at the origin. Therefore its Milnor fibre is homotopy equivalent to a bouquet of
spheres of dimension $(n_g-1)$.
\noindent {\bf 2.} For a germ $f:(\CC^n,0)\to (\CC,0)$, let ${\rm Re\,}f:(\CC^n,0)\to (\RR,0)$ be its real
part and let $V_{{\rm Re\,}f}=({\rm Re\,}f)^{-1}(\varepsilon)\cap B_{\delta}^{2n}$
($0<\varepsilon\ll\delta$) be its ``real Milnor fibre''.
In~\cite{Ruan_etal} the space $\calH_g$ is defined as
$\calH_g:=H^{n_g}(B_{\delta}^{2n},V_{{\rm Re\,}f^g};\CC)^G$. However this space
is canonically isomorphic to $H^{n_g-1}(V_{f^g};\CC)^G$ (with the conventions for $n_g=0,1$ in the
definition above).
\noindent {\bf 3.} One can consider the {\em quantum homology group} of the pair $(f,G)$ defined as
\begin{equation}\label{quantum_homology}
\bigoplus_{g\in G} H_{n_g-1}(V_{f^g};\CC)^G\,,
\end{equation}
where each summand on the right hand side is the $G$-invariant part of the
(middle) homology group $H_{n_g-1}(V_{f^g};\CC)^G= H_{n_g-1}(V_{f^g}/G;\CC)$ of the Milnor fibre
of the restriction of $f$ to the fixed point set of $g$. The majority of the constructions
below are valid both for the quantum cohomology group and for the homology one.
\end{remarks}
Our aim is to define orbifold versions of the Milnor lattice and of the intersection form.
In singularity theory the intersection form is traditionally considered on the vanishing
homology group. Therefore below we shall mostly consider the quantum homology group
using the same notations $\calH_{f,G}$ and $\calH_g$. (In fact in \cite{Ruan_etal}
the description of the quantum cohomology group and of some structure on it starts
from the discussion of the corresponding homology group.)
\section{Orbifold monodromy operator} \label{sect:mono}
For a germ $f:(\CC^n,0)\to(\CC,0)$ with an isolated critical point at the origin the (classical)
monodromy transformation is a map $\varphi_f$ from the Milnor fibre
$V_f$ to itself
induced by rotating the (noncritical) value $\varepsilon$ around zero counterclockwise (see, e.g.,
\cite{AGV2}). If the function $f$ is quasihomogeneous, i.e., if there exist positive integers
$w_1$, \dots, $w_n$ and $d$ such that
$f(\lambda^{w_1}z_1, \ldots, \lambda^{w_n}z_n)=\lambda^df(z_1, \ldots, z_n)$
for $\lambda\in\CC$, this transformation can be defined by
\[
\varphi_f(z_1, \ldots, z_n)=(\exp(2\pi i \cdot w_1/d)z_1, \ldots , \exp(2\pi i \cdot w_n/d)z_n).
\]
This transformation is an element of the symmetry group of the function $f$. In the
FJRW theory it is usually denoted by $J$. The fix point set $(\CC)^J$ of the element $J$ is zero-dimensional.
Therefore the corresponding summand in (\ref{quantum_cohomology}) is one-dimensional. A generator of this
summand represents the unit element of the corresponding cohomological field theory.
The action of the monodromy transformation on the vanishing homology group of the singularity
$f$ is called the (classical) monodromy operator and will be denoted by $\varphi_f$ as well.
If the germ $f$ is invariant with respect to an action of a finite abelian group $G$ on $\CC^n$,
the classical monodromy transformation $\varphi_f$ can be assumed to be $G$-equivariant.
This implies that it preserves the fixed point sets of subgroups of $G$ in the Milnor fibre,
i.e., for a subgroup $K\subset G$ (in particular, for $K=\langle g\rangle$, $g\in G$), the
map $\varphi_f$ sends $V_{f^K}=V_f\cap (\CC^n)^K$ to itself and also induces a map
$\widehat{\varphi}_{f^K}$ from the quotient space $V_{f^K}/G$ to itself. The actions of these
maps on the homology groups $\calH_g$ define a map from the quantum homology group
$\calH_{f,G}$ to itself.
The orbifold monodromy zeta function of
a pair $(f,G)$ consisting of a germ of a function $(\CC^n,0)\to(\CC,0)$ (not necessarily
non-degenerate, i.e.\ with an isolated critical point at the origin) and a finite group $G$ of its symmetries
(not necessarily abelian)
was defined in \cite{EG-Edinburgh}.
We recall the definition for an abelian group $G$. The usual monodromy zeta function
of the transformation ${\widehat{\varphi}}_{f^{\langle g\rangle}}$ is defined by
\begin{equation*}
\zeta_{{\widehat{\varphi}}_{f^{\langle g\rangle}}}(t)=\prod\limits_{q\ge 0}
\left(\det({\rm id}-t\cdot {\widehat{\varphi}}_{f^{\langle g\rangle}}^{*}
{\rm \raisebox{-0.5ex}{$\vert$}}{}_{H^q_c(V_{f^{\langle g\rangle}}/G;\RR)})\right)^{(-1)^q}\,
\end{equation*}
The definition of the orbifold monodromy zeta function is inspired by the notion of the orbifold spectrum (see, e.g., \cite{BH2, ET}). For an
element $g \in G$ acting on $\CC^n$ by
\[ g(z_1, \ldots , z_n)=(\exp(2 \pi i \cdot r_1)z_1, \ldots , \exp(2\pi i \cdot r_n)z_n)
\]
where $0 \leq r_j < 1$, $j=1, \ldots , n$, its {\em age} (or fermion shift number) \cite{Ito-Reid, Zaslow} is
\[ {\rm age}(g) = \sum_{j=1}^n r_j \in \QQ_{\geq 0}.
\]
The map $g \mapsto \exp(2\pi i \cdot {\rm age}(g))$ defines a character
$\alpha_{\rm age} \in G^\ast={\rm Hom}(G,\CC^\ast)$.
For an abelian group $G$, the orbifold monodromy zeta function $ \zeta^{{\rm orb}}_{f,G}(t)$ is given by the following equation
\begin{equation*}
\zeta^{{\rm orb}}_{f,G}(t)= \prod\limits_{g\in G}
\left(\zeta_{{\widehat{\varphi}}_{f^{\langle g\rangle}}}(\exp(-2\pi i\,{\rm age\,}(g))t)\right)\,.
\end{equation*}
The reduced orbifold monodromy zeta function $\overline{\zeta}^{{\rm orb}}_{f,G}(t)$ is defined by
$$
\overline{\zeta}^{{\rm orb}}_{f,G}(t)=
\zeta^{{\rm orb}}_{f,G}(t)\left/\prod\limits_{g\in G}(1-\exp(-2\pi i\,{\rm age\,}(g))t) \right..
$$
It was shown that the reduced orbifold monodromy zeta functions of
Berglund--H\"ubsch--Henningson dual pairs (not necessarily non-degenerate ones) either coincide or are inverse
to each other depending on the number $n$ of variables. If the function $f$ has an isolated critical point
at the origin, the reduced monodromy zeta function coincides with the {\em characteristic polynomial} of
the pair $(f,G)$ defined in \cite{ET} (which is not always a polynomial).
The definition of the orbifold monodromy zeta function leads to the following definition.
\begin{definition} The {\em orbifold monodromy operator} $\varphi_{f,G}$ on the quantum (co)homology group
$\calH_{f,G}$ is the direct sum of the operators $\alpha_{\rm age}(g) \cdot \widehat{\varphi}_{f^g}$ on
$\calH_g$.
\end{definition}
One can show that the reduced monodromy zeta function $\overline{\zeta}^{\rm orb}_{f,G}(t)$ coincides with
the zeta function of the orbifold monodromy operator $\varphi_{f,G}$ .
The orbifold monodromy operator does not preserve the natural lattice $H_{n_g-1}(V_{f^g};\ZZ)^G$
in $\calH_g$ (and, in general, no lattice in it at all). A lattice in $\calH_{f,G}$ preserved by
the orbifold monodromy operator $\varphi_{f,G}$ will be described below.
\section{Group rings} \label{sect:rings}
Let $R$ be a commutative ring with unity (usually either the field $\CC$ of complex numbers
or the ring $\ZZ$ of integers).
For a finite abelian group $K$, let $R[K]$ be the corresponding group ring.
It is a free $R$-module with the basis $\{e_g\}$ whose elements correspond to the elements $g$ of the
group $K$. Let $K^*={\rm Hom\,}(K,\CC^*)$ be the group of characters of the group $K$.
As an abstract group $K^*$ is isomorphic to $K$, but not in a canonical way.
The space $\CC[K]$ carries a natural representation of the group $K$ defined by $he_g=e_{hg}$ for
$h\in K$. A change of the basis permits to identify $\CC[K]$ as a vector space (not as a ring) with the vector space
$\CC[K^*]$. Namely, one should define the new basis $\e_{\alpha}$, $\alpha\in K^*$, by
\begin{equation}\label{change}
\e_{\alpha}:=\sum_{g\in K}\left(\alpha(g)\right)^{-1}e_g\,.
\end{equation}
In the other direction one has
\begin{equation}\label{change_back}
e_g=\frac{1}{\vert K\vert}\sum_{\alpha\in K^*}\left(\alpha(g)\right)\e_{\alpha}\,.
\end{equation}
For a character $\beta:K\to\CC^*$, let $\psi_{\beta}$ be the linear map from $\CC[K]$ to $\CC[K]$
defined by
$$
\psi_{\beta}(e_g)=\beta(g)e_g\,.
$$
One has
$$
\psi_{\beta}(\e_{\alpha})=\sum_{g\in K}\left(\alpha(g)\right)^{-1}\beta(g)e_g=
\sum_{g\in K}\left(\alpha(g)(\beta(g))^{-1}\right)^{-1}e_g=\e_{\alpha\beta^{-1}}\,.
$$
This implies that the map $\psi_{\beta}$ preserves the lattice $\ZZ[K^*]\subset\CC[K^*]$.
For a subgroup $H$ of $K$ one has a natural map from $K^*$ to $H^*$: the restriction of characters.
This map is epimorphic. It induces a ring epimorphism $r^K_H:\CC[K^*]\to \CC[H^*]$.
\begin{lemma} \label{lem:Ker}
The kernel ${\rm Ker\,} r^K_H$ of the homomorphism $r^K_H$ coincides with the subspace of $\CC[K]$ generated by the basis
elements $e_g$ with $g\in K\setminus H$.
\end{lemma}\label{lemma1}
\begin{proof}
Let $A_H$ be the kernel of the natural map $K^*\to H^*$. From (\ref{change_back}) one has
$$
r^K_H e_g=\frac{1}{\vert K\vert}\sum_{\beta\in H^*}
\left(\sum_{\alpha\in K^*:
\alpha_{\vert H}=\beta}\alpha(g)\right)\widehat{e}_\beta=
\frac{1}{\vert K\vert}\sum_{\beta\in H^*}
\left(\widehat{\beta}(g) \sum_{\alpha\in A_H}\alpha(g)\right)\widehat{e}_\beta\,,
$$
where $\widehat{\beta}$ is an element of $K^*$ such that $\widehat{\beta}_{\vert H}=\beta$.
The element $g$ as an element of $\left(K^*\right)^*=K$ defines a non-trivial character on
the subgroup $A_H$. Therefore $\sum\limits_{\alpha\in A_H}\alpha(g)=0$ and thus $r^K_H e_g=0$.
This shows that $\langle e_g:g\in K \setminus H\rangle\subset {\rm Ker\,}r^K_H$. On the other hand the dimensions of
these spaces are equal to $\vert K\vert-\vert H\vert$.
\end{proof}
\begin{remark}
If $K$ is the maximal group $G_f$ of diagonal symmetries of an invertible polynomial $f$ and
$H$ ia a subgroup of $K$, then the dual group $\widetilde{H}$ in the Berglund--H\"ubsch--Henningson dual
pair $(\widetilde{f},\widetilde{H})$ is the kernel of the map $K^*\to H^*$ indicated above.
Pay attention that ${\rm Ker\,} r^K_H$ is not isomorphic to $\CC[\widetilde{H}]$. (In particular,
the dimension of the latter one is equal to $\vert K\vert/\vert H\vert$.)
\end{remark}
The map $\psi_{\beta}$ described above preserves the lattice
$\ZZ[K^*]\cap {\rm Ker\,} r^K_H\subset {\rm Ker\,} r^K_H$.
\section{Orbifold Milnor lattice} \label{sect:Milnor}
Here we define a lattice in the quantum homology group $\calH_{f,G}$
which can be considered as an orbifold version of the Milnor lattice.
Let $G$ be a finite abelian group acting faithfully on $(\CC^n,0)$. Let $f:(\CC^n,0)\to(\CC,0)$ be a germ of a $G$-invariant
holomorphic function with an isolated critical point at the origin.
For a point $x\in \CC^n$, let
$G_x=\{g\in G\, | \, gx=x\}$ be the isotropy subgroup of $x$. One has $(\CC^n)^K=\{x\in \CC^n \,|\,
G_x\supset K\}$. Let ${\rm Iso\,}G$ be the set of the subgroups of $G$ which are isotropy subgroups
of some points (i.e., $K\in {\rm Iso\,}G$ iff $\exists x\in\CC^n:G_x=K$).
Let $K$ be a subgroup of $G$ belonging to ${\rm Iso\,}G$. Let $E_K\subset \CC[K^*]$ be the
intersection of the kernels of the maps $r^K_H$ for all $H \in {\rm Iso\,}G$ such that
$H\varsubsetneq K$. Lemma~\ref{lem:Ker} implies that the subspace $E_K$ is generated by
all the basis elements $e_g$ of $\CC[K]$ with
$$
g\in {{\stackrel{\circ}{K}}}:=
K\setminus\bigcup\limits_{\begin{array}{c}H\in{\rm Iso\,}G,\atop H\varsubsetneq K\end{array}}H\,.
$$
There is a natural lattice $\ZZ[K^*]$ in $\CC[K^*]$. Its intersection with the subspace $E_K$
gives a lattice there. Using it one can define a lattice in the quantum homology group
$\calH_{f,G}$ in the way described below. The definition of the lattice $\ZZ[K^*]\cap E_K$
(and thus of the corresponding lattice in $\calH_{f,G}$) does not take into account the ages
of the elements of the group $G$ although they constitute an important part of the quantum
singularity theory. Moreover symmetric bilinear forms on $\calH_{f,G}$ which can be constructed
in a somewhat natural way and which could be considered as orbifold analogues of the (symmetric)
intersection form on the vanishing homology group of a singularity appear to be not integral
or at least not even on the corresponding lattice.
Therefore we consider another lattice in $\CC[K^*]$ and thus in $\calH_{f,G}$.
Let $\alpha_K\in K^*$ be the restriction of $\alpha_{\rm age}\in G^*$ to $K$ (in particular,
$\alpha_G=\alpha_{\rm age}$) and let $p_K$ be the order of $\alpha_K$. Let $\ZZ^{(p_K)}[K^*]$
be the sublattice of $\ZZ[K^*]$ defined by
$$
\ZZ^{(p_K)}[K^*]=\left\{\sum\limits_{\alpha\in K^*}m_{\alpha}\widehat{e}_{\alpha}\, \left| \,
\forall \beta\in K^*:\ \sum_{j=1}^{p_K} m_{\beta\alpha_K^j}
\ \ {\text{is divisible by}}\ \ p_K\right.\right\}\,.
$$
(In other words the sublattice $\ZZ^{(p_K)}[K^*]$ is the kernel of the natural map
$\ZZ[K^*]\to\ZZ_{p_K}[K^*]$.)
Let $E_K^{\ZZ}:=E_K\cap \ZZ^{(p_K)}[K^*]$.
Let
$$
\calH_K:=H_{n_K-1}(V_{f^K};\CC)^G\,.
$$
The space $\calH_K$ contains the natural lattice
$\calH_K^{\ZZ}=H_{n_K-1}(V_{f^K};\ZZ)^G$.
By definition
$$
\calH_{f,G}=\bigoplus_{g\in G} \calH_g\,.
$$
All the summands on the right hand side are of the form $\calH_K$ for $K\in {\rm Iso\,}G$.
The space $\calH_K$ appears as the summand $\calH_g$ if and only if $g\in{\stackrel{\circ}{K}}$.
Thus one has
$$
\calH_{f,G}=\bigoplus_{K\in {\rm Iso\,}G}\bigoplus_{g\in {\stackrel{\circ}{K}}} \calH_K\,.
$$
Therefore
\begin{equation} \label{eq:qhg}
\calH_{f,G}=\bigoplus_{K\in {\rm Iso\,}G}E_K\otimes \calH_K\,.
\end{equation}
The tensor product
of two complex vector spaces with distinguished lattices contains a natural lattice as well.
Therefore the quantum homology group $\calH_{f,G}$ contains the natural lattice
$\Lambda_{f,G}=\bigoplus\limits_{K\in {\rm Iso\,}G} E_K^{\ZZ}\otimes \calH_K^{\ZZ}$.
Let us recall that $\psi_{\alpha_{K}}$ is a map from $\CC[K]$ to itself sending the basis
element $e_g$ to $\alpha_{K}(g)e_g$. The orbifold monodromy operator is
$$
\varphi_{f,G} = \bigoplus_{K\in {\rm Iso\,}G}\psi_{\alpha_{K}}\otimes \widehat{\varphi}_{f^K},
$$
where $\widehat{\varphi}_{f^K}$ is the map $\calH_K\to\calH_K$ induced by the classical monodromy operator.
Since $(\psi_{\alpha_{K}})_{\vert E_K}$ and $\widehat{\varphi}_{f^K}$ preserve the corresponding lattices
in $E_K$ and in $\calH_K$, the orbifold monodromy operator $\varphi_{f,G}$ preserves the lattice
$\Lambda_{f, G}$. Thus we have proved the following statement.
\begin{theorem}\label{theo-lattice}
There exists a well defined lattice $\Lambda_{f, G}$ in the quantum homology group
$\calH_{f, G}$ invariant with respect to the orbifold monodromy operator $\varphi_{f,G}$.
\end{theorem}
\begin{definition}
The lattice $\Lambda_{f, G}$ in $\calH_{f, G}$ will be called the {\em orbifold Milnor lattice}
of the pair $(f,G)$.
\end{definition}
\section{Orbifold Seifert form} \label{sect:Seifert}
An essential aim of this paper is to define an analogue of the intersection form on the
quantum homology group (or on the orbifold Milnor lattice). To describe properties of
the intersection form and its relations with the monodromy transformation, it is useful
to use the Seifert form (or the so-called variation operator), see, e.g., \cite{AGV2}.
For a germ $f:(\CC^n,0) \to (\CC,0)$ of a holomorphic function with an isolated critical point
at the origin, the Seifert form is a (non-degenerate) bilinear form on the vanishing homology
group (the Milnor lattice) $H_{n-1}(V_f;\ZZ)$ ($V_f$ is the Milnor fibre of $f$), or, in other words,
a linear map $L:H_{n-1}(V_f;\ZZ) \to (H_{n-1}(V_f;\ZZ))^\ast$ (see, e.g., \cite{AGV2}).
In general, this form is neither symmetric nor skew-symmetric. (The group $(H_{n-1}(V_f;\ZZ))^\ast$
dual to $H_{n-1}(V_f;\ZZ)$ is isomorphic to the relative homology group
$H_{n-1}(V_f, \partial V_f;\ZZ)$ or to the cohomology group $H^{n-1}(V_f;\ZZ)$.)
In a so-called distinguished basis of the Milnor lattice $H_{n-1}(V_f;\ZZ)$ (and the dual
basis of $(H_{n-1}(V_f;\ZZ))^\ast$), the matrix of the operator $L$ is an upper triangular matrix with
the diagonal elements equal to $(-1)^{\frac{n(n+1)}{2}}$. The intersection form $S$ on
$H_{n-1}(V_f;\ZZ)$ is equal to
\begin{equation} \label{eq:S}
S=-L+ (-1)^n L^T,
\end{equation}
where $L^T$ is the transposed form. The monodromy operator
$\varphi_f : H_{n-1}(V_f;\ZZ) \to H_{n-1}(V_f;\ZZ)$ is given by the equation
\begin{equation} \label{eq:mon}
\varphi_f = (-1)^n L^{-1}L^T.
\end{equation}
An important advantage of the Seifert form (compared with the intersection form) is the formula
for the Seifert form of the Sebastiani-Thom (``direct'') sum of singularities.
If $f_1:(\CC^m,0) \to (\CC,0)$ and $f_2:(\CC^n,0) \to (\CC,0)$ are two germs with isolated
critical points at the origin then the vanishing homology group $H_{m+n-1}(V_{f_1 \oplus f_2}; \ZZ)$
of the germ $f_1 \oplus f_2: (\CC^{m+n},0) \to (\CC,0)$,
$(f_1 \oplus f_2)(\underline{x}, \underline{y}) = f_1(\underline{x})+f_2(\underline{y})$,
is canonically isomorphic to the tensor product $H_{m-1}(V_{f_1};\ZZ) \otimes H_{n-1}(V_{f_2};\ZZ)$
of the corresponding vanishing homology groups. One has (see, e.g., \cite[Theorem~2.10]{AGV2})
\begin{equation} \label{eq:Thom}
L_{f_1 \oplus f_2} = (-1)^{mn} L_{f_1} \otimes L_{f_2}.
\end{equation}
A translation of this property in terms of the intersection form can be formulated in a
reasonable way only in a distinguished basis and is given by somewhat involved formulae.
If a finite group $G$ acts on $(\CC^n,0)$ and $f: (\CC^n,0) \to (\CC,0)$ is a $G$-invariant
germ with an isolated critical point at the origin, the monodromy operator preserves the
$G$-invariant part of the vanishing homology group, the restriction $L^G$ of the Seifert form
to the $G$-invariant part is non-degenerate and it is related with the restriction
of the monodromy operator and of the intersection form by the same equations as (\ref{eq:S})
and (\ref{eq:mon}). If $f_1$ and $f_2$ are two germs as above and $f_1$ is $G$-invariant
with respect to an action of a finite group $G$ on $\CC^m$, then $f_1 \oplus f_2$ is
$G$-invariant with respect to the $G$-action on $\CC^m \oplus \CC^n$ which is the trivial
extension of the one on $\CC^m$. One has
$H_{m+n-1}(V_{f_1 \oplus f_2}; \ZZ)^G \simeq
H_{m-1}(V_{f_1};\ZZ)^G \otimes H_{n-1}(V_{f_2};\ZZ)$ and
\begin{equation} \label{eq:Thom_equi}
L^G_{f_1 \oplus f_2} = (-1)^{mn} L^G_{f_1} \otimes L_{f_2}\,.
\end{equation}
Here we define a ``Seifert form'' (an integer valued bilinear form) on the orbifold
Milnor lattice $\Lambda_{f,G}$. For that we shall identify $\CC[K^*]\otimes \calH_K$,
$K\in {\rm Iso\,}G$, with a direct sum of vanishing homology groups of certain singularities
so that the orbifold monodromy operator on it becomes the direct sum of (the restrictions of)
the corresponding classical monodromy operators. The direct sum of the corresponding
Seifert forms on the summands
gives a bilinear form on $\CC[K^*]\otimes \calH_K$ with integer values on the lattice
$\ZZ^{(p_K)}[K^*]\otimes \calH_K^{\ZZ}$.
The direct sum over all subgroups
$K\in {\rm Iso\,}G$ of the restrictions of these forms to $E_K\otimes \calH_K$ gives a bilinear
form on $\calH_{f,G}$ with integer values on the orbifold Milnor lattice $\Lambda_{f,G}$ which will be
called the orbifold Seifert form.
The space $\CC[K^*]$ has a decomposition into parts corresponding to the orbits
of the multiplication by $\alpha_K$, i.e., to the elements of $K^*/\langle\alpha_K\rangle$:
$$
\CC[K^*]=\bigoplus_{\gamma\in K^*/\langle\alpha_K\rangle}\langle\widehat{e}_{\alpha}:
[\alpha]=\gamma\rangle\,.
$$
Each summand $B_{\gamma}=\langle\widehat{e}_{\alpha}: [\alpha]=\gamma\rangle$ on the right hand side
can be represented as the direct sum of two subspaces:
$B_{\gamma,1}\cong\CC$ generated by the element $\sum\limits_{[\alpha]=\gamma}\widehat{e}_{\alpha}$
and $B_{\gamma,2}$ consisting of the elements
$\sum\limits_{[\alpha]=\gamma}c_{\alpha}\widehat{e}_{\alpha}$ such that $\sum c_{\alpha}=0$.
The subspaces $B_{\gamma,1}\cong\CC$ and $B_{\gamma,2}$ contain natural lattices:
$\langle\sum\limits_{[\alpha]=\gamma}\widehat{e}_{\alpha}\rangle$ and
$\ZZ^{(p_K)}[K^*]\cap B_{\gamma,2}=\ZZ[K^*]\cap B_{\gamma,2}$ respectively.
If $p_k=1$, then $B_{\gamma,2}=0$. If $p_k \geq 2$, then the space $B_{\gamma,2}$ with the
lattice $\ZZ^{(p_K)}[K^*]\cap B_{\gamma,2}=\ZZ[K^*]\cap B_{\gamma,2}$
in it can be identified with the vanishing homology group of the $A_{p_K-1}$-singularity
with the Milnor lattice in it in the following way.
The $A_{p_K-1}$-singularity is defined by the function $u^{p_K}$. The Milnor fibre
$V_{u^{p_K}}=\{u^{p_K}=1\}$ of it consists of the points $u_j=\exp{(-2\pi i (j-1)/p_K)}$,
$j=1,\ldots,p_K$. A (distinguished) basis of the Milnor lattice in the vanishing homology
group $\widetilde{H}_0(V_{u^{p_K}})$ is formed by the vanishing cycles
$\Delta_1=u_1-u_2$, $\Delta_2=u_2-u_3$, \dots, $\Delta_{p_K-1}=u_{p_K-1}-u_{p_K}$
(see, e.g., \cite[Theorem 2.5]{AGV2}, where one has a small misprint (a wrong sign)
in the formula for $u_j$; in the initial Russian version the sign is correct).
Sometimes it is convenient to consider also the vanishing cycle $\Delta_{p_K}=u_{p_K}-u_{1}$
keeping in mind that $\sum\limits_{j=1}^{p_k}\Delta_{j}=0$.
The Seifert operator $L$ is defined by $L\Delta_1=-\Delta_1^*$,
$L\Delta_j=-\Delta_j^*+\Delta_{j-1}^*$ for $j>1$, where $\Delta_1^*$, $\Delta_2^*$, \dots, $\Delta_{p_K-1}^*$
is the dual basis in the dual lattice. (The monodromy transformation of the function $u^{p_K}$
permutes the points $u_j$ cyclically sending $u_j$ to $u_{j-1}$ and therefore sending
the cycle $\Delta_j$ to $\Delta_{j-1}$ for $j>1$ and the cycle $\Delta_1$ to
$\Delta_{p_K}=-\sum\limits_{j=1}^{p_K-1}\Delta_j$.) Let us consider the following (integer) basis of
$B_{\gamma,2}$. Let $\alpha\in K^*$ be an arbitrary representative of $\gamma$.
Then the set $\delta_1=\widehat{e}_{\alpha_K\alpha}-\widehat{e}_{\alpha}$,
$\delta_2=\widehat{e}_{\alpha_K^2\alpha}-\widehat{e}_{\alpha_K\alpha}$, \dots,
$\delta_{p_K-1}=\widehat{e}_{\alpha_K^{p_K-1}\alpha}-\widehat{e}_{\alpha_K^{p_K-2}\alpha}$
is a basis of $\ZZ^{(p_K)}[K^*]\cap B_{\gamma,2}$. We identify $B_{\gamma,2}$ with the
vanishing homology group $\widetilde{H}_0(V_{u^{p_K}})$ of the $A_{p_K-1}$-singularity
by identifying the vanishing cycles $\Delta_j$ with the basis elements $\delta_j$.
This identification respects the corresponding lattices.
\begin{remark}
It is important to note that the bilinear form on $B_{\gamma,2}$ defined through this
identification by the Seifert form on the $A_{p_K-1}$-singularity and therefore also analogues
of the Seifert forms defined below do not depend on the choice of the representative
$\alpha$ in an orbit $\gamma$.
\end{remark}
To define an analogue $\widehat{L}_K$ of the Seifert form on $\CC[K^*]\otimes\calH_K$,
we shall define an analogue $\widehat{\ell}_K$ of the Seifert form on $\CC[K^*]$.
Let $\widehat{\ell}_K$ be the direct sum of the Seifert forms $\ell_{\gamma,2}=L_{u^{p_K}}$ of the
$A_{p_K-1}$-singularity on $B_{\gamma, 2}$ and the form $\ell_{\gamma,1}$ on $B_{\gamma,1}\cong\CC$
defined by
$$
\ell_{\gamma,1}\left(\sum\limits_{\alpha:[\alpha]=\gamma}\widehat{e}_{\alpha},
\sum\limits_{\alpha:[\alpha]=\gamma}\widehat{e}_{\alpha} \right)
=-1\,.
$$
Let $I:\CC[K^*]\to\CC[K^*]$ be the operator which is the identity on $B_{\gamma, 2}$ and minus
the identity on $B_{\gamma, 1}$ for each $\gamma$.
\begin{proposition}
The bilinear form $\widehat{\ell}_K$ is integer valued on the lattice $E_K^{\ZZ}$ and possesses the property
\begin{equation}\label{psi_alpha}
\psi_{\alpha_K}=-I \, \widehat{\ell}_K^{-1} \widehat{\ell}_K^T\,.
\end{equation}
\end{proposition}
\begin{proof}
The lattice $\ZZ^{(p_K)}[K^\ast]$ is the direct sum of the lattices $\ZZ[K^*]\cap B_{\gamma,2}$ and
$\ZZ[K^*]\cap B_{\gamma,1}$. The form $\widehat{\ell}_K$ is integer valued on them and they are orthogonal
to each other with respect to $\widehat{\ell}_K$. Therefore $\widehat{\ell}_K$ is integer valued on
$\ZZ^{(p_K)}[K^*]\supset E_K^{\ZZ}$.
The restriction of the operator $\psi_{\alpha_K}$ to $B_{\gamma,2}$ coincides with the
monodromy operator of the $A_{p_K-1}$-singularity and therefore is equal to $-\widehat{\ell}_K^{-1}\widehat{\ell}_K^T$.
The restriction of $\psi_{\alpha_K}$ to the (one-dimensional) space $B_{\gamma,1}$ is the identity and thus coincides with
$\widehat{\ell}_K^{-1}\widehat{\ell}_K^T$.
\end{proof}
\begin{remark}
The presence of the operator $I$ in Equation~(\ref{psi_alpha}) is related with the fact
that, in some sense, $B_{\gamma,2}$ and $B_{\gamma,1}$ are identified with the vanishing
homology groups of functions with different numbers of variables: $1$ and $0\mod 4$
respectively. Stabilization of a function means addition of the sum of squares
of new variables. If one of two functions is right-equivalent to the stabilization of the other one,
the functions are called stably equivalent. Topological properties of stably equivalent functions
are closely related. Their Milnor lattices can be identified. Via these identification, the Seifert forms
of stably equivalent functions may differ, but only by a sign. Moreover, they are 4-periodic: if the numbers of
variables of functions have the same residue modulo 4, their Seifert forms coincide. The monodromy
transformations of stably equivalent functions are 2-periodic. Thus from the topological point of view
only the residue of the number of variables modulo 4 matters.
The orbifold monodromy operator acts on $B_{\gamma,2}$ and on $B_{\gamma,1}$ as on the vanishing homology
group of the $A_{p_K-1}$-singularity of 1 variable and the $A_1$-singularity of 0 variables respectively.
\end{remark}
\begin{theorem}\label{orb-Z}
The restriction $\ell_K$ of the bilinear form $\widehat{\ell}_K$ to $E_K$ is non-degenerate
and possesses the property
\begin{equation}
\psi_{\alpha_K}=-I \, \ell_K^{-1}\ell_K^T.
\end{equation}
\end{theorem}
\begin{proof}
Let $H\in {\rm Iso\,}G$, $H\varsubsetneq K$, and let $A_H$ be the kernel of the natural map
$K^*\to H^*$. The subgroup $A_H\subset K^*$ acts on $\CC[K^*]$ by permutations of the basis
elements $\widehat{e}_{\alpha}$. Let $\CC[K^*]=\CC[K^*]^{A_H}\oplus\CC[K^*]^{\overline{A_H}}$
be the decomposition of $\CC[K^*]$ into the invariant part $\CC[K^*]^{A_H}$ and the
``non-invariant part'' $\CC[K^*]^{\overline{A_H}}$, i.e., the sum of all the parts corresponding
to non-trivial representations of $A_H$. For $\beta\in H^*$, the group $A_H$ acts on the subspace
$\langle\widehat{e}_{\alpha}: \alpha_{\vert H}=\beta\rangle\subset \CC[K^*]$ by the regular
representation. This representation is the direct sum of the (one-dimensional) invariant part
generated by the element $\sum\limits_{\alpha:\alpha_{\vert H}=\beta}\widehat{e}_{\alpha}$
and the non-invariant part consisting of all the linear combinations of the elements
$\widehat{e}_{\alpha}$, $\alpha_{\vert H}=\beta$, with the sum of the coefficients equal to zero. The non-invariant part
coincides with the kernel of the restriction to
$\langle\widehat{e}_{\alpha}: \alpha_{\vert H}=\beta\rangle$ of the map $r^K_H:\CC[K^*]\to\CC[H^*]$.
This implies that ${\rm Ker\,}r^K_H=\CC[K^*]^{\overline{A_H}}$.
For all subgroups $H\in {\rm Iso\,}G$ such that $H\varsubsetneq K$, the actions of the subgroups
$A_H\subset K^*$ commute and the space $\CC[K^*]$ decomposes into the parts corresponding to
different representations of these subgroups. The subspace $E_K$ is the intersection of the
subspaces $\CC[K^*]^{\overline{A_H}}$ for all $H\in {\rm Iso\,}G$, $H\varsubsetneq K$.
This means that it is the (direct) sum of all the parts on which the representations
of all the groups $A_H$ are non-trivial.
The operator $\psi_{\alpha_K}$ commutes with these actions and the bilinear form
$\widehat{\ell}_K$ is invariant with respect to them. This implies that these parts are
orthogonal to each other with respect to the bilinear form $\widehat{\ell}_K$, the restriction
of $\widehat{\ell}_K$ to each of these parts is non-degenerate and satisfies the relation
(\ref{psi_alpha}). This implies the statement.
\end{proof}
\begin{definition}
The Seifert form $\widehat{L}_K$ on $\CC[K^*]\otimes\calH_K$ is
$$
\widehat{L}_K=(-1)^{n_K}\widehat{\ell}_K\otimes L^G_{f^K}
$$
(cf.~(\ref{eq:Thom_equi}): the Seifert form $\widehat{L}_K$ is defined as the Seifert form of
the Sebastiani-Thom sum of functions of one variable and of $n_K$ variables respectively).
\end{definition}
The representation of $\CC[K^*]$ in the form
$$
\bigoplus_{\gamma\in K^*/\langle\alpha_K\rangle}\left(\widetilde{H}_0(V_{u^{p_K}};\CC)\oplus\CC\right)
$$
gives an isomorphism between $\CC[K^*]\otimes\calH_K$ and
\begin{eqnarray*}
& & \bigoplus_{\gamma\in K^*/\langle\alpha_K\rangle}\left(\widetilde{H}_0(V_{u^{p_K}};\CC)\otimes\calH_K\oplus\calH_K\right) \\
& = & \bigoplus_{\gamma\in K^*/\langle\alpha_K\rangle}\left(H_{n_K}(V_{f^K+u^{p_K}};\CC)^G\oplus H_{n_K-1}(V_{f^K};\CC)^G\right)\,.
\end{eqnarray*}
The form $\widehat{L}_K$ is the direct sum of the Seifert forms on the first summands and the Seifert forms on the second summands stabilized to
the same number of variables.
Equation~(\ref{psi_alpha}) implies that
\begin{equation}\label{L-mono}
\psi_{\alpha_K}\otimes \widehat{\varphi}_{f^K}=-I \, \widehat{L}_K^{-1}\widehat{L}_K^{T}.
\end{equation}
\begin{theorem}
The restriction $L_K$ of the form $\widehat{L}_K$ to $E_K\otimes\calH_K$ is a non-degenerate bilinear form such that
$$
(\psi_{\alpha_K}\otimes \widehat{\varphi}_{f^K})_{\vert E_K\otimes\calH_K}=-I \, L_K^{-1}L_K^{T}.
$$
\end{theorem}
\begin{proof}
This is a direct consequence of Equation~(\ref{L-mono}) and Theorem~\ref{orb-Z}.
\end{proof}
\begin{definition} The {\em orbifold Seifert form} $L_{f,G}$ on the quantum homology group $\calH_{f,G} = \bigoplus E_K \otimes \calH_K$ is the direct sum of the forms $L_K$ on $E_K \otimes \calH_K$.
\end{definition}
\section{Intersection forms on the orbifold Milnor lattice} \label{sect:inter}
Here we define bilinear forms on the quantum homology group which are analogues of the intersection forms
on the vanishing homology groups of singularities. The intersection form on the vanishing homology group
of a singularity is either symmetric or skew-symmetric depending on the number of variables. To each singularity
one also associates a symmetric form which is the intersection form of its stabilization with an odd number of
variables. (A tradition of singularity theory is to consider the stabilization with the number of variables
equal to $3\mod 4$.) The symmetric intersection form appears to be a more important invariant of singularities than the non-symmetric one.
For a germ $f:(\CC^n,0)\to(\CC,0)$ the intersection form $S(\cdot,\cdot)$ on the vanishing homology group
$H_{n-1}(V_f,\CC)$ is defined by the Seifert form. Namely, one has
$S=-L+(-1)^nL^T$. This inspires the following definition.
\begin{definition}
The {\em mixed intersection form} on the quantum homology group $\calH_{f,G}$ is defined by
$$
S^{\rm mix}_{f,G}=\bigoplus_{K\in {\rm Iso\,}G}\left(-L_K+(-1)^{n_K}L_K^T\right)\,.
$$
\end{definition}
It is an integer valued bilinear form on $\calH^{\ZZ}_{f,G}$ (symmetric or skew symmetric on the summands in (\ref{eq:qhg})).
Essentially (up to sign) there are two natural symmetric bilinear forms on the quantum homology group $\calH_{f,G}$.
One of them is obtained by the stabilization of each summand to a function of $3\mod 4$ variables. The other one
(more natural from our point of view) is obtained by the stabilizations to $3\mod 4$ variables for $n_K$ odd
and to $1\mod 4$ variables for $n_K$ even respectively (or vice versa).
\begin{definition}
The {\em orbifold intersection form} on the quantum homology group $\calH_{f,G}$ is
$$
S^{\rm orb}_{f,G}=\bigoplus_{K\in {\rm Iso\,}G}(-1)^{\frac{n_K(n_K+1)}{2}}\left(-L_K-L_K^T\right)\,.
$$
The {\em quantum intersection form} on $\calH_{f,G}$ is
$$
S^{\rm qua}_{f,G}=\bigoplus_{K\in {\rm Iso\,}G}(-1)^{\frac{(n_K-2)(n_K+1)}{2}}\left(-L_K-L_K^T\right)\,.
$$
\end{definition}
The reason for these names is the following. The quantum intersection form is ``predominantly negative''.
This means that it is induced by intersection forms of singularities with the self-intersection numbers
of the vanishing cycles equal to $(-2)$. The orbifold intersection form is ``predominantly negative''
on the summands with $n_K$ odd and is ``predominantly positive'' (i.e., induced by intersection forms
with the self-intersection numbers of the vanishing cycles equal to $(+2)$) on the summands with $n_K$ even.
In the second case the signature of this form is more related to the orbifold Euler characteristic,
whence in the first case the signature is more related to the rank of the quantum homology group.
Summarizing the facts from Section~\ref{sect:Seifert}, we have
\begin{proposition}
The bilinear forms $S^{\rm mix}_{f,G}$, $S^{\rm orb}_{f,G}$ and $S^{\rm qua}_{f,G}$ are integer valued forms on
$\calH_{f,G}^{\ZZ}$. The forms $S^{\rm orb}_{f,G}$ and $S^{\rm qua}_{f,G}$ are symmetric and even.
\end{proposition}
\begin{remark}
It is interesting to understand a relation of the defined bilinear forms with the bilinear pairing
considered in the FJRW-theory: \cite[page 38]{Ruan_etal}. However, a direct relation is unclear.
The pairing in the FJRW-theory is well defined only if the group $G$ contains the exponential
grading operator $J$, whereas the definitions of the pairings introduced here do not require
additional conditions on the group $G$. The pairing in the FJRW-theory is defined through pairings
on the summands $\calH_g$ of the quantum (co)homology group \cite[Definition~3.1.1]{Ruan_etal}.
In fact the pairing on $\calH_g$ is nothing else but the Seifert form of the germ $f^g$ restricted to
the subspace of the vanishing
homology group of $f^g$ invariant with respect to $G$. The Seifert form itself is not symmetric,
however its restriction to the subspace invariant with respect to the classical monodromy operator $J$ is.
\end{remark}
\section{Invertible polynomials} \label{sect:inv}
In this section, we compute the orbifold Milnor lattice for some examples. These examples are chosen in the class of so called invertible polynomials. An invertible polynomial is a quasihomogeneous polynomial with the number of monomials equal to the number of variables. We consider an invertible polynomial $f$ and a subgroup $G$ of the group $G_f$ of diagonal symmetries of $f$. A description of properties of invertible polynomials and of their symmetry groups can be found, e.g., in \cite{Krawitz, KS, EGT}. In particular, for a pair $(f,G)$ consisting of a (non-degenerate) invertible polynomial $f$ and a subgroup $G$ of its symmetry group $G_f$, one can consider the (Berglund--H\"ubsch--Henningson) dual pair $(\widetilde{f}, \widetilde{G})$. It is an interesting problem to compare the orbifold Milnor lattices for dual pairs. We shall examine some examples.
The first result is that the quantum cohomology groups of dual pairs have the same rank. This was shown in \cite[Theorem~1.1]{Krawitz} for a pair $(f,G)$, where $G$ is an admissible group, i.e., a group containing the exponential grading operator $J$. Here we give a proof for an arbitrary group $G$. For this purpose we adapt certain results of \cite{EGT}.
The quantum cohomology group $\calH_{f,G}$ is the direct sum of the subspaces $\calH_{f,G,0}$ and $\calH_{f,G,1}$ where
\[
\calH_{f,G,i} = \bigoplus_{g \in G, \atop n_g \equiv i \, {\rm mod} \, 2} \calH_g \quad \mbox{for } i=0,1.
\]
A ($\QQ \times \QQ$)-grading on these spaces was defined in \cite[Equations~(2.2),(2.3)]{EGT} in terms of the mixed Hodge structures on the vanishing cohomology groups and the ages of elements of $G$.
It is the same one as the bigrading considered in \cite[Remark 3.2.4]{Ruan_etal}.
For the next definition compare \cite[Equation~(2.4)]{EGT}.
\begin{definition}
The {\em E\,$^i$-function} of the pair $(f,G)$, $i=0,1$, is
\begin{equation}
E^i(f,G)(t,\bar{t}):=\displaystyle\sum_{p,q\in\QQ} {\rm dim}_\CC (\calH_{f,G,i})^{p,q} \cdot t^{p-\frac{n}{2}}{\bar{t}}^{q-\frac{n}{2}}.
\end{equation}
\end{definition}
For the E-function considered in \cite{EGT} one has
\[
E(f,G)(t,\bar{t})= E^0(f,G)(t,\bar{t}) - E^1(f,G)(t,\bar{t}).
\]
One has the following relations between the ${\rm E}^i$-functions of a pair $(f,G)$ and the dual pair $(\widetilde{f}, \widetilde{G})$, which are refined versions of \cite[Theorem~9]{EGT}.
\begin{theorem} For $n$ even one has
\[ E^i(f,G)(t,\bar{t}) = E^i(\widetilde{f},\widetilde{G})(t^{-1},\bar{t}) \quad \mbox{for } i=0,1.
\]
For $n$ odd one has
\begin{eqnarray*}
E^0(f,G)(t,\bar{t}) & = & E^1(\widetilde{f},\widetilde{G})(t^{-1},\bar{t}), \\
E^1(f,G)(t,\bar{t}) & = & E^0(\widetilde{f},\widetilde{G})(t^{-1},\bar{t}).
\end{eqnarray*}
\end{theorem}
\begin{proof} Let us define
\[
E'(f,G)(t,\bar{t}) := E^0(f,G)(t,\bar{t}) + E^1(f,G)(t,\bar{t}).
\]
The arguments used in \cite{EGT} imply that the function $E'(f,G)(t,\bar{t})$ is given by the equations (2.8) and (2.9) without the sign $(-1)^{n_g}$ in the latter one. The computations presented in \cite[Section~4]{EGT} give the following equation for $E'(f,G)(t,\bar{t})$ (cf.\ \cite[Proposition~14]{EGT})
\begin{equation}
E(f,G)(t, \bar{t}) = \sum_{(g, \widetilde{g}) \in G \times \widetilde{G}} \widehat{m}_{g,\widetilde{g}} (t \bar{t})^{{\rm age}(g)-\frac{n-n_g}{2}} \left( \frac{\bar{t}}{t} \right)^{{\rm age}(\widetilde{g})-\frac{n-n_{\widetilde{g}}}{2}},
\end{equation}
where the numbers $\widehat{m}_{g,\widetilde{g}}$ are defined in \cite{EGT}. Since $\widehat{m}_{\widetilde{g},g}=\widehat{m}_{g,\widetilde{g}}$, this equation implies the statement.
\end{proof}
\begin{corollary}
One has
\[
\dim \calH_{f,G} = \dim \calH_{\widetilde{f},\widetilde{G}}
\]
$($and therefore ${\rm rk} \, \Lambda_{f,G} = {\rm rk} \, \Lambda_{\widetilde{f},\widetilde{G}}$$)$.
\end{corollary}
\begin{example} \label{Ex0}
We consider the invertible polynomial $f(x,y)=x^2y+y^5$ with its maximal group of symmetries $G=G_f$. Here $G_f$ is the group generated by the exponential grading operator $(\exp (2 \pi i)2/5), \exp((2 \pi i)1/5))$ and $(-1,0)$. The orbifold Milnor lattice is the direct sum
\[ \Lambda_{f,G} = \left( \bigoplus_{K \in {\rm Iso}\, G \atop K \neq \{ {\rm id} \}} \bigoplus_{g\in {\stackrel{\circ}{K}}} \calH_K^\ZZ \right) \oplus H_1(V_f; \ZZ)^G.
\]
We first show that $H_1(V_f; \ZZ)^G=0$. For this we consider a suitable real morsification of the function $f$ and the distinguished basis of vanishing cycles obtained by the method of N.~A'Campo and the second author from it (see, e.g., \cite[Section~4.1]{AGV2}). The corresponding Coxeter-Dynkin diagram is shown in Fig.~\ref{Fig0}.
\begin{figure}
$$
\xymatrix{ *{\bullet} \ar@{-}[dr] \ar@{}^{2}[r] & & & &\\
& *{\oplus} \ar@{-}[r] \ar@{}^{5}[d] & *{\bullet} \ar@{-}[r] \ar@{}^{3}[d] & *{\oplus} \ar@{-}[r] \ar@{}^{6}[d] & *{\bullet} \ar@{}^{4}[d] \\
*{\bullet} \ar@{-}[ur] \ar@{}_{1}[r] & & & &
}
$$
\caption{Coxeter-Dynkin diagram of $f(x,y)=x^2y+y^5$} \label{Fig0}
\end{figure}
One can easily see that the elements of $H_1(V_f;\ZZ)$ which are invariant under the monodromy operator $J$ are linear combinations of the vanishing cycles corresponding to the saddle points (indicated by $\bullet$) such that the sum of the coefficients along the boundary of a region (indicated by $\oplus$ or $\ominus$) to which they are connected is equal to zero. Here these elements are generated by the basic elements
$1-2$ and $1-3+4$.
None of them is invariant under the transformation $(x,y) \mapsto (-x,y)$ which corresponds to the reflexion at the horizontal axis.
By our definition, the remaining part of the orbifold Milnor lattice $\Lambda_{f,G_f}$ with the orbifold intersection form is isomorphic to $A_1 \oplus A_4 \oplus A_4$. On the other hand, the dual polynomial $\widetilde{f}(x,y)=x^2+xy^5$ defines an $A_9$-singularity with the dual group $\widetilde{G}_f= \{ {\rm id}\}$.
\end{example}
We now examine two examples of Krawitz \cite[3.2]{Krawitz}.
\begin{example} \label{ExA}
We consider the invertible polynomial $f(x,y)=x^3y+xy^5$ of loop type (see \cite{KS}) with the group $G$ generated by the exponential grading operator $J=(\exp (2 \pi i)2/7), \exp((2 \pi i)1/7))$. The polynomial is self-dual and $\widetilde{G}=\langle (-1,-1) \rangle$. The orbifold Milnor lattice with respect to $G$ has a natural splitting
\[ \Lambda_{f,G} = {{\stackrel{\circ}{\Lambda}}} \oplus H_1(V_f; \ZZ)^G \mbox{ where } {{\stackrel{\circ}{\Lambda}}} =\bigoplus_{g \in G\setminus \{ {\rm id} \} } \calH_g^\ZZ.
\]
The lattice ${{\stackrel{\circ}{\Lambda}}}$ is isomorphic to $A_6$. In order to compute the invariant part $H_1(V_f; \ZZ)^G$ of the usual Milnor lattice of $f$, we proceed as in Example~\ref{Ex0}. A Coxeter-Dynkin diagram with respect to a suitable real morsification of the function $f$ is given by Fig.~\ref{FigA}.
\begin{figure}
$$
\xymatrix{
*{\bullet} \ar@{-}[r] \ar@{}_{5}[d] & *{\ominus} \ar@{-}[r] \ar@{-}[d] \ar@{--}[dr] \ar@{}_{1}[d] & *{\bullet} \ar@{-}[d] \ar@{-}[r] \ar@{}^{7}[d] & *{\ominus} \ar@{-}[r] \ar@{-}[d] \ar@{--}[dl] \ar@{--}[dr] \ar@{}^{2}[d] &*{\bullet} \ar@{-}[d] \ar@{}^{10}[d] & & \\
& *{\bullet} \ar@{-}[r] \ar@{}_{6}[d] & *{\oplus} \ar@{-}[r] \ar@{-}[r] \ar@{-}[d] \ar@{--}[dr] \ar@{}_{14}[d] & *{\bullet} \ar@{-}[r] \ar@{-}[d] \ar@{}_{9}[d] & *{\oplus} \ar@{-}[r] \ar@{-}[d] \ar@{--}[dl]\ar@{--}[dr] \ar@{}_{15}[d] & *{\bullet} \ar@{-}[d] \ar@{}_{12}[d] & \\
& & *{\bullet} \ar@{-}[r] \ar@{}_{8}[d] & *{\ominus} \ar@{-}[r] \ar@{}_{3}[d] & *{\bullet} \ar@{-}[r] \ar@{}_{11}[d] & *{\ominus} \ar@{-}[r] \ar@{}_{4}[d] & *{\bullet} \ar@{}_{13}[d] \\
& & & & & & }
$$
\caption{Coxeter-Dynkin diagram of $f(x,y)=x^3y+xy^5$} \label{FigA}
\end{figure}
A basis of the subspace of invariant cycles is given by the elements
\[ 5-6+8-11+12, 5-7+9-11+13, 6-7+10-12+13 \]
The matrix for the Seifert form $L^G$ with respect to this basis is given by
\[ \left( \begin{array}{ccc} -5 & -2 & 2 \\
-2 & -5 & -2\\
2 & -2 & -5
\end{array} \right)
\]
It has determinant $-49$.
Now we consider the dual group $\widetilde{G}$. The orbifold Milnor lattice is
\[ \Lambda_{\widetilde{f},\widetilde{G}}= A_1 \oplus H_1(V_{\widetilde{f}}; \ZZ)^{\widetilde{G}}.
\]
The group $\widetilde{G}$ acts on the diagram of Fig.~\ref{FigA} by reflection at the central vertex 9. A basis of the subspace of $\widetilde{G}$-invariant cycles is given by
\[ 1+4, 2+3, 5+13, 6+12, 7+11, 8+10, 9, 14+15.
\]
The matrix of the Seifert form $L^{\widetilde{G}}$ with respect to this basis is given by
\[ \left( \begin{array}{cccccccc} -2 & 0 & -2 & -2 & -2 & 0 & 0 & 2\\
0 & -2 & 0 & 0 & -2 & -2 & -2 & 2\\
0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -2 & 0 & 0 & 0 & -2\\
0 & 0 & 0 & 0 & -2 & 0 & 0 & -2\\
0 & 0 & 0 & 0 & 0 & -2 & 0 & -2\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & -2\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -2
\end{array} \right)
\]
It has determinant 128.
\end{example}
\begin{example} \label{ExB}
We consider the invertible polynomial $f(x,y)=x^3y+y^4$ of chain type \cite{KS}, again with the group G generated by the exponential grading operator which is in this case $J=(\exp (2 \pi i)/4), \exp((2 \pi i)/4))$. The orbifold Milnor lattice of the pair $(f,G)$ again has a natural splitting
\[ \Lambda_{f,G} = {{\stackrel{\circ}{\Lambda}}} \oplus H_1(V_f; \ZZ)^G \mbox{ where } {{\stackrel{\circ}{\Lambda}}} =\bigoplus_{g \in G\setminus \{ {\rm id} \} } \calH_g^\ZZ.
\]
The lattice ${{\stackrel{\circ}{\Lambda}}}$ is in this case isomorphic to $A_1 \oplus A_1 \oplus A_1$. In order to compute the invariant part $H_1(V_f; \ZZ)^G$ of the usual Milnor lattice of $f$, we again proceed as in Example~\ref{Ex0}. A Coxeter-Dynkin diagram with respect to a suitable real morsification of the function $f$ is given in Fig.~\ref{FigB}.
\begin{figure}
$$
\xymatrix{ & & *{\bullet} \ar@{-}[d] \ar@{}_{2}[d] & & \\
& *{\bullet} \ar@{-}[r] \ar@{-}[d] \ar@{}_{3}[d] & *{\ominus} \ar@{-}[r] \ar@{-}[d] \ar@{--}[dr] \ar@{--}[dl] \ar@{}_{1}[d] & *{\bullet} \ar@{-}[d] \ar@{}_{4}[d] & \\
*{\bullet} \ar@{-}[r] \ar@{}_{6}[d] & *{\oplus} \ar@{-}[r] \ar@{}_{8}[d] & *{\bullet} \ar@{-}[r] \ar@{}_{5}[d] & *{\oplus} \ar@{-}[r] \ar@{}^{9}[d] &*{\bullet} \ar@{}^{7}[d] \\
& & & & }
$$
\caption{Coxeter-Dynkin diagram of $f(x,y)=x^3y+y^4$} \label{FigB}
\end{figure}
A basis of the subspace of invariant cycles is given by the elements
\[ 2-3+6, 2-4+7, 2-5+6 \]
The matrix of the Seifert form $L^G$ with respect to this basis is given by
\[ \left( \begin{array}{ccc} -3 & -1 & 1 \\
-1 & -3 & -1\\
1 & -1 & -3
\end{array} \right)
\]
It has determinant $-16$.
Now we consider the dual pair $(\widetilde{f},\widetilde{G})$.
The dual polynomial is $\widetilde{f}(x,y)=x^3+xy^4$ and the dual group is $\widetilde{G}=\langle (\exp (2 \pi i)1/3), \exp((2 \pi i)2/3)) \rangle$. The orbifold Milnor lattice is
\[ \Lambda_{\widetilde{f},\widetilde{G}}= A_2 \oplus H_1(V_{\widetilde{f}}; \ZZ)^{\widetilde{G}}.
\]
In order to compute the Seifert form $L^{\widetilde{G}}$ on $H_1(V_{\widetilde{f}}; \ZZ)^{\widetilde{G}}$, we work with the polynomial $h(x,y)=x^3+y^6$ which is in the same $\mu$-constant equivariant stratum as $\widetilde{f}(x,y)=x^3+xy^4$. A distinguished basis of vanishing cycles for this polynomial can be computed by the method of A.~M.~Gabrielov \cite{Gab}. It is obtained as follows: Let $e_1,e_2$ be a distinguished basis of vanishing cycles for $A_2$ ($x^3$) and $f_1, \ldots , f_5$ a distinguished basis of vanishing cycles for
$A_5$ ($y^6$). Then
\begin{equation} \label{eq:Gab}
\gamma_{ij} = e_i \otimes f_j
\end{equation}
is a distinguished basis of vanishing cycles for $h$. We extend these sets by $e_3:=-(e_1+e_2)$ and $f_6:=-(f_1 + \cdots + f_5)$. We extend the definition (\ref{eq:Gab}) to $i=3$ and $j=6$ as well. One has
\[ L(\gamma_{ij}, \gamma_{ij})=-1, L(\gamma_{ij}, \gamma_{i+1,j})=L(\gamma_{ij}, \gamma_{i,j+1})=1, L(\gamma_{ij}, \gamma_{i+1,j+1})=-1
\]
and $L(\gamma_{ij}, \gamma_{i'j'})=0$ otherwise, where $i+1=1$ for $i=3$ and $j+1=1$ for $j=6$. Then one can compute that the following cycles form a basis of the subspace of $\widetilde{G}$-invariant cycles:
\begin{eqnarray*}
b_{22} & = & \gamma_{22} + \gamma_{36}+\gamma_{14}, \\
b_{23} & = & \gamma_{23}+\gamma_{31}+\gamma_{15}, \\
b_{24} & = & \gamma_{24}+\gamma_{32}+\gamma_{16},\\
\delta & = & \gamma_{12}+\gamma_{13}+\gamma_{14}+\gamma_{15}+\gamma_{16}+\gamma_{22}+\gamma_{23}+\gamma_{24}+\gamma_{32}.
\end{eqnarray*}
The matrix of the Seifert form $L^{\widetilde{G}}$ with respect to this basis is given by
\[ \left( \begin{array}{cccc} -3 & 3 & 3 & 3 \\
0 & -3 & 3 & 0\\
0 & 0 & -3 & -3 \\
0 & 0 & 0 & -1
\end{array} \right) .
\]
It has determinant $27$.
\end{example}
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new City guide.
You will find the new feature in the “More” menu inside the Facebook app:
and, just to give you an idea it looks like this:
The functionality is meant to suggest and even send notifications (that’s still to come, though) about places your friends have visited, places the locals go, upcoming events and popular attractions… in short, anything you might have an interest in while visiting a city.
If you are traveling alone, don’t worry, it will also display names of friends that have shown an interest in the same events as you, so you can easily find your wingman/woman to come along with you. You can also click on each of your friend’s pictures to see what that did when they were there.
It hasn’t rolled out for everyone yet (one of my phones is in French, and there is no sign of it in the app for example), and will not display all the cities in the world (right now, I can only see the 6 that I posted above) – but the feature has some potential considering how much travel information we feed in the app.
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Jack Layton Ferry Terminal design competition begins
Designs are available for viewing online or in the main rotunda at City Hall
More than a million visitors flow through the Jack Layton Ferry Terminal each year on their way to or from the Toronto Islands. But it's now more than 40 years since the terminal was built, leaving the space overdue for a makeover.
This week five design teams are presenting for public input their proposals to remake the space. All five plans are available for viewing here on Waterfront Toronto's website, but they can also be seen in person in the City Hall rotunda until Friday.
The goal of the design competition is to create a unifying and inspiring master plan for the terminal and its surrounding area that can be phased in over time.
Christopher Glaisek of Waterfront Toronto spoke about the project in an interview Tuesday on CBC Radio's Metro Morning.
He said the design competition is the first step toward addressing a number of long-standing shortcomings of the current terminal, including:
- Poor traffic flow for passengers, particularly on busy summer days when demand greatly exceeds the terminal's capacity.
- An improved aesthetic to move away from the current "cattle pen" feel and chain-link fence perimeter. There are also few places for passengers to sit while they're waiting for the next ferry. "You feel like you're in prison and you're about to go to the most liberating place in the city," said Glaisek.
- Better integration with both the lake and Bay Street. Right now the terminal isn't easy to locate, a problem for tourists.
So what's the process?
Glaisek said he's hoping the design competition and public input will create a "big-picture idea" of what the new ferry terminal should look like. There is currently no money allocated to the project, but he said if support forms around some solid design ideas, that could spur forward the terminal's makeover.
"Until you have an idea, it's hard to get money," he said.
The five designs will be evaluated by a master jury and recommendations will go to Waterfront Toronto and begin a master planning process.
Here's a quick look at the five designs:
Design team: Stoss Landscape Urbanism (Boston) + nARCHITECTS (New York City) + ZAS Architects (Toronto)
Key feature: A star-shaped terminal building with a ceiling that allows natural light to filter in.
Design team: Clement Blanchet Architecture (Paris) + Batlle i Roig (Barcelona) + RVTR (Toronto and Ann Arbor) + Scott Torrance Landscape Architect Inc. (Toronto)
Key feature: A raised walkway into the terminal that will connect the site with the bottom of Bay Street.
Name: Civic Canopy
Design team: Diller Scofidio+Renfro (New York City) + architectsAlliance (Toronto) + Hood Design (Emeryville, CA)
Key feature: An eye-catching open-air roof that resembles a wave.
Name: Harbour Landing
Design team: KPMB Architects (Toronto), West 8 (Rotterdam), Greenberg Consultants (Toronto)
Key feature: Terminal building would be located under an undulating natural roof covered with grass.
Name: Jack Layton Ferry Terminal and Harbour Square Park
Design Team: Quadrangle Architects (Toronto), aLLDesign (London), Janet Rosenberg & Studio (Toronto)
Key feature: In addition to a large terminal building overlooking the ferry slips, this design calls for an elevated pathway across.
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With they way they kept baiting each other, intentionally hurting each other emotionally blah-blah-blah, I think they will continue to do so even after the story ends.
No amount of I-love-yous can stop that.
I'm not complaining about above, I'm actually looking forward to reading more verbal sparing scenes (though some makes me go oh Hugh... grow up!). And the making up later. ☺
Some of my fav lines from A Virtuous Lady:
In normal circumstances, he would never have given her a second glance, his Briony. He repeated the name under his breath. Even her name had the power to bewitch him.I need to give myself an english name pronto. Oh wait, does 'Fancy' count?
"Listen to me Miss Briony Langland. The man you give yourself to will be me or you will give yourself to no man."Be still my heart.
His longing for Briony had long since surpassed the mere physical desire to satiate his lust. He could not live without her. He wanted her to share his life. It was as simple as that.Want.
"Good God, Briony, the fact that I have chosen to bind myself to one woman irrevocably should prove something to you!"Oh yes. Good God, Briony.
And the whole part where they got caught in a storm and then Hugh was holding Briony and comforting her? All this thoughts then just wow-ed me. ♥
****
Don't you just hate the situation where you just finished a book and then have to choose between ∞ books to read next?
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We desert dwellers all like to joke about the heat that descends, during certain months, on the valley like a blazing comet on a direct impact path. The sweltering temperatures are part of the advanced level of swimming pool architecture you see all over the Las Vegas metroplex. We have to beat the heat somehow.
These days, though, our thermostat isn’t the only example of mercury rising in Las Vegas. Our favorite planned community, Summerlin, is fanning the flames of our luxury real estate market. Our team at The Dream Home Specialist brings you this, our Las Vegas Luxury Real Estate Report.
Summerlin Basics: What’s Not to Love?
An ironic aspect of Summerlin is that the village was vacant for its first 30 years of existence. Summerlin Parkway was nicknamed “The road to nowhere” by Las Vegas residents. It’s difficult to imagine the now 100,000-citizen community as an empty expanse leading up to the Red Rock Canyon.
Building commenced in Summerlin when The Meadows School broke ground in 1988 and hasn’t looked back since. Technically, Summerlin is a community, not a separate town, even though it feels far removed from your typical Las Vegas experience.
Within the borders of this award-winning planned development, you’ll find 20 or so villages, all with a unique set of amenities. Linking these clusters is a trail system of about 150 total miles. When Howard Hughes began planning out Summerlin, he wanted each collection of homes to have a neighborhood park (150 total today) and a “sidewalk” that joined each segment. Those footpaths evolved to the trail system we see today.
Adding to the appeal of Summerlin are the educational opportunities found within the village. More than two-dozen public and private schools encourage learning, athletics, music, art and life skills to Summerlin’s youngest residents.
Summerlin: Something For Everyone
If Summerlin ever adopts a new tagline, the phrase “If you can’t find it in Summerlin, it doesn’t exist” should be a contender. Using luxury real estate as an example, within Summerlin, the village of Sun City, The Ridges, and The Summit Club (opening soon). Sun City Summerlin is an age-restricted development; The Ridges is luxury golf course community with custom homes on large parcels, and The Summit is the most luxurious development yet.
You can golf on one of the village’s nine golf courses, including The Bear’s Best. Jack Nicklaus designed Bear’s Best by picking out his favorite hole layouts from his library of plans at Nicklaus Designs, Inc. This collection brings accolades from professionals and casual players, both. Surrounding the course are custom home sites in The Ridges development.
In Sun City, a 55-year and older age-restricted development, you’ll find three golf courses, a movie theater, 14 tennis courts, indoor and outdoor swimming pools, and three restaurants. Have dinner and a movie without ever leaving your extended neighborhood.
Throughout Summerlin, the Fun Continues
I’m going to throw in the extensive list of cultural and interest-related activities as part of Summerlin’s hot real estate status. Townspeople do not skimp when it comes to filling the event calendar. Every month you’ll see dozens of happenings within the community.
And the business owners fully support community efforts. Rachel’s Kitchen near Trails Village shopping center displays works from the local artisans and craftspeople of Summerlin, including students.
Social and activity clubs add to your options, and then there are art and culture venues to explore. These are a small sampling of what’s going on around the village:
- Sun City Summerlin Art Club
- Nevada Ballet Theatre
- The Starbright Theater
- Red Rock Resort
- The Summerlin Writers and Poets Club
- Discovery Children’s Museum
- The Women’s Club of Summerlin
- Red Rock Canyon National Conservation Area Visitor Center
- The Smith Center for the Performing Arts
- Ward 2 Walkers fitness group
- West Side Photo Club
- Las Vegas Crafters Guild
- The Rotary Club of Las Vegas-Summerlin
- The Fresh52 farmers market
Village Square’s monthly festival featuring vintage cars is a throwback to bygone days of straight pipes and big block engines. Tour de Summerlin, in its 15th year, attracts hundreds of cyclists to test their endurance on 40, 60, and 80-mile rides.
Summerlin is Calling You for Further Exploration!
All of this luxury real estate and you’ll enjoy proximity to Las Vegas, too. You’re living the best of both worlds, and if you aren’t yet contact us today!
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TITLE: A generalized K- theory via generalized idempotents
QUESTION [6 upvotes]: Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach algebra or a ring with $a^{n}=a$.
Can the 3 equivalent relations, Murray-Von Neumann, similarity and homotopy on 2-idempotents be generalized to n-idempotents,for arbitrary $n>2$? Does this processes gives us a useful and new type of K theory?
We know that "Vector bundles" are the topological analogy of 2-idempotents. Now what is a topological analogy for generalized idempotents?
REPLY [14 votes]: Let $E_n(A)$ be the set of $n$-idempotents in $A$, and let $u_1,\dotsc,u_n$ be the elements of $E_n(\mathbb{C})$. Let $E'_n(A)$ be the set of $n$-tuples $e_1,\dotsc,e_n\in E_2(A)$ with $e_ie_j=0$ for $i\neq j$, and $\sum_ie_i=1$. Define $f\colon E'_n(A)\to E_n(A)$ by $f(e_1,\dotsc,e_n)=\sum_iu_ie_i$. Then it is not hard to see that $f$ is bijective. Thus, $E_n(A)$ does not really tell you anything that is not already determined by $E_2(A)$.
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Strategic Integration) 744-2989
1435 Cleveland Ave, Loveland, CO
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249 South College Ave., Fort Collins, CO
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1294 Main St, Windsor, CO
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search results
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JS XV6
What's this?
This is an online demo of our xv6 port to GAIA, the CPU we designed.
JS porting is done by compiling our CPU simulator with Emscripten.
JS XV6 is very very fast! In fact, it is faster than the native binary.
Xv6 is a simple UNIX like OS by MIT. If you want to know more about
xv6 or our porting project, see links below.
Command Examples
- ls
See what commands you can run. PATH is fixed to "/".
- 2048
A popular game clone. You can use arrow keys or hjkl keys.
- sl
Let's correct your typo with the famous steam locomotive.
...
Links
@nullpo_head
- This JS simulator repository
- XV6-GAIA repository
- Original simulator repository
- Original xv6
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I met with the PTA team this week to explore how things are going, what we could do differently and what we would like to keep the same. They are a great team who work so hard and are happy to do as much as they can for the whole school, I know many try to help with events but the load is still too big. Also in response to feedback from Fayres and the feeling that you have to give lots, we are going to explore doing things differently to make the average £3,000 a year.
There will no longer be a Christmas or Summer Fayre. These are being swapped for a termly raffle:
There will continue to be:
New Ideas:
The team are exploring fun, family events that will be a great way to raise funds. The first of these is a Colour Run, parents pay a £7.50 entrance fee per child or adult who come dressed for the event with protective glasses or goggles on. Each person is given a large cup of powder paint (eco safe) and whilst they are running they can throw and receive paint. You can then purchase more for £2 a cup. It looks crazy fun. We would look to hold this on the 26th June instead of a Fayre. We would also sell refreshments, drinks and ice creams, at this event.
In order to find out if you would like to join in with this, you will receive a Marvellous Me, high 5 back if you say yes, ignore if you say no.
If you have any great ideas for fun family fundraising events, please let the team or your Class Rep know.
With World Book Day coming soon, 6th March 2020. Sarah Maude proposed a great idea of having a pop up shop. Anyone who has an old, grown out of, don’t want anymore, costume; you can donate them to the school in the next week. Then the week after half term we will put them out for sale on the playground, with a cost of £2 per costume. It is a great way of helping the environment and the purse.
If you have any costumes to donate, please bring them to the school office by next Friday.
With the warmer winters we are going to start swimming after half term, it will begin on the 2nd March so time to dig out costumes, trunks, hats, towels and suitable footwear. Below is a timetable of when each class will swim.
On Thursday 13th 3.15-3.45pm and Friday 14th 8.30-9am the children will be sharing all their wonderful work please come along and enjoy your child’s achievements.
Year 6 have been very busy working on a Chocolate Enterprise Project as part of PSHE, they have chosen lots of exciting names for their chocolates and will be selling them for 50p bag after school on Friday 14th February. All proceeds will be donated to our Year 6 Class Charity - Shooting Star Chase Hospice.
On sale Friday 14th February on the playground. Paper bags of sweets are 50p, however if you bring your own reusable container you’ll get a slightly larger scoop of sweets for your 50p! Sweets will be on sale on both playgrounds. There will be a sugar free alternative.
Do you have a 2, 3, or 4year old? Come along to our Open Morning/Afternoon to meet the staff and see our fantastic facilities. There is no need to book just pop along at your preferred time.
Wednesday 12th February 9.30-11am & 1.30-2.30 pm
Due to family reasons Mrs Cooper is currently out of the office. The other members of the office team are available to help. If your child is being collected by someone else, please let the classroom teachers know when you drop off in the morning.
Points are being awarded this week for trackers, any child who achieves 5,000 earns 100 points for their team, 10,000 = 300 points and there is an extra 100 points for every additional 1,000 steps. Get walking!
Next week is Week 3 –.
Just add Ecosia to google chrome and while you search the web on your phone or tablet,they use the 100% of the profit made from your searches to plant trees.
They have already planted over 3 million trees! If you would like to find out more and follow their progress click on the link below.
If you have not yet finalised your payment for this terms’ Funzone, please ensure you return this to the office or pay online as soon as possible. Without payment you will be unable to book for next term.
Booking for next terms’ Funzone is already open (form attached)..
Kind regards,
Mrs Curtis
School Dinners - Week 4
Reminder
There is an inset on Monday 24th February. Children return from Half Term on Tuesday 25th February.
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\begin{document}
\title[Artin-Schreier curves]{An improvement of the Hasse-Weil bound for Artin-Schreier curves via cyclotomic function fields}
\thanks{}
\author{Liming Ma}\address{School of Mathematical Sciences, University of Science and Technology of China, Hefei China
230026}\email{lmma20@ustc.edu.cn}
\author{Chaoping Xing} \address{School of Electronic Information and Electric Engineering, Shanghai Jiao Tong University,
China 200240}\email{xingcp@sjtu.edu.cn}
\maketitle
\begin{abstract}
The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to improve the Hasse-Weil bound for Artin-Schreier curves.
In this paper, we focus on the number of rational places of the Artin-Schreier curve defined by $y^p-y=f(x)$ over the finite field $\F_q$ of characteristic $p$, where $f(x)$ is a polynomial in $\F_q[x]$. Our road map for attacking this problem works as follows. We first show that the function field $E_f:=\F_q(x,y)$ of the Artin-Schreier curve $y^p-y=f(x)$ is a subfield of some cyclotomic function field. We then make use of the class field theory to prove that the number of points of the curve is upper bounded by a function of a minimum distance of a linear code. By analyzing the minimum distance of this linear code, we can improve the Hasse-Weil bound and Serre bound for Artin-Schreier curves.
\end{abstract}
\section{Introduction}
Throughout this paper, let $p$ be a prime and let $q=p^m$ for some integer $m\ge 1$. Let $f(x)\in\F_q[x]$ be a polynomial and consider the set
\[Z_f:=\{\Ga\in\F_q:\; \Tr(f(\Ga))=0\},\]
where $\Tr$ is the trace function from $\F_q$ to $\F_p$. For simplicity, throughout this paper, we assume that the degree of $f(x)$ is $r$ with $r<p$.
Then the Weil bound shows that the cardinality $|Z_f|$ is bounded by the following inequality
\begin{equation}\label{eq:1.1}
\left||Z_f|-q^{m-1}\right|\le \frac{(r-1)(p-1)}{p}\sqrt{q}.
\end{equation}
A good upper bound on $|Z_f|$ has found various applications such as exponential sums \cite{Ca80, KL11, La91, MK93, WW16}, BCH codes in coding theory \cite{AL94, GV95,SV94,Wo88}
and nonlinearity of boolean functions \cite{AM14, Ca10}, etc. Let $N_f$ be the number of rational points of the Artin-Schreier curve:
\begin{equation}\label{eq:1.1a}
y^p-y=f(x).
\end{equation}
Based on the relation $N_f=1+p|Z_f|$,
the bound \eqref{eq:1.1} is derived from the following Hasse-Weil bound for Artin-Schreier curves
\begin{equation}\label{eq:1.2}
|N_f-1-q|\le (r-1)(p-1)\sqrt{q}.\end{equation}
In literatures, there are various improvements on the Hasse-Weil bound \eqref{eq:1.2} for Artin-Schreier curves.
Firstly Serre improved the Hasse-Weil bound for arbitrary curves of genus $g$ over $\F_q$ to the following bound
\begin{equation}\label{eq:1.3}
|N-q-1|\le g\lfloor 2\sqrt{q}\rfloor\end{equation}
by analyzing the property of algebraic integers of its L-polynomial \cite[Theorem 5.3.1]{St09}, where $N$ stands for the number of points of the curve. The bound \eqref{eq:1.3} is called the Serre bound. If we apply the Serre bound to the Artin-Schreier curve
$y^p-y=f(x)$, we get the following bound
\begin{equation}\label{eq:1.4}
|N_f-q-1|\le \frac{(r-1)(p-1)}2\times \lfloor 2\sqrt{q}\rfloor.\end{equation}
In 1993, Moreno and Moreno provided an improvement to the Hasse-Weil bound for Artin-Schreier curves by the property of exact division \cite{MM93}.
Kaufman and Lovett proved that the Hasse-Weil bound can be improved for Artin-Schreier curves $y^p-y=f(x)$ with $f(x)=g(x)+h(x)$, where $g(x)$ is a polynomial with $\deg (g(x))\ll \sqrt{q}$ and $h(x)$ is a sparse polynomial of arbitrary degree with bounded weight in \cite{KL11}.
Rojas-Leon and Wan showed that an extra $\sqrt{p}$ can be removed for this family of curves if $p$ is very large compared with polynomial degree of $f(x)$ and $\log_p q$ in \cite{RW11}.
Cramer and Xing showed that the property of exact division related to the number of rational places can be improved by using L-polynomials of algebraic curves with the Hasse-Witt invariant $0$ in \cite{CX17}.
In this paper, we will focus on the upper bound of the number of rational points of Artin-Schreier curves as well. However, our approach is completely different from those in all previous papers. More precisely speaking, our approach is divided into the following steps.
{\it Step 1:} We first assume that an Artin Schreier curve has at least two rational points. In this case, there is a rational place corresponding to $x-\Ga$ of $\F_q(x)$ that splits completely in the function field $E_f$ of an Artin Schreier curve. Thus, we can show that the function field $E_f$ of an Artin-Schreier curve is a subfield of the cyclotomic function field $K_r=K(\Lambda_{T^{r+1}})$ with modulo $T^{r+1}$, where $K=\F_q(x)$ and $T=(x-\Ga)^{-1}$. As $K_r/K$ is a Galois extension with Galois group $\Gal(K_r/K)\simeq\F_q^*\times \F_p^{mr}$ and $\Gal(E_f/K)\simeq \F_p$, where $E_f=\F_q(x,y)$ with $y^p-y=f(x)$, $E_f$ is in fact a subfield of $F$, where $F$ is the subfield of $K_r$ fixed by $\F_q^*$. Thus, $\Gal(F/K)\simeq\F_p^{mr}$ can be viewed as an $\F_p$-vector space of dimension $mr$ and the Galois group $\Gal(F/E_f)$ is a subspace of $\Gal(F/K)$ of dimension $mr-1$.
{\it Step 2:} To see whether $x=\Ga-\Gb$ has $p$ solutions in the equation $y^p-y=f(x)$ for $\Gb\in \F_q^*$, by the class field theory it is equivalent to checking if $x-(\Ga-\Gb)$ or equivalently $1+\Gb T$, belongs to the group $\Gal(F/E_f)$. Let $\F_q^*=\{\Ga_1,\dots,\Ga_{q-1}\}$ and a subset $I\subseteq[q-1]$, the space spanned by $\{1+\Ga_i T\}_{i\in I}$ has $\F_p$-rank $rm$ if and only if the matrix $(\Ga_i^j)_{1\le j\le r, i\in I}$ has $\F_p$-rank $rm$. This gives an upper bound on $N_f$ that is a function of the minimum distance of the linear code over $\F_p$ generated by the matrix $(\Ga_i^j)_{1\le j\le r, 1\le i\le q-1}$, where $\Ga_i^j$ are viewed column vectors via an isomorphism between $\F_q$ and $\F_p^m$.
{\it Step 3:} By investigating the minimum distance of the above linear code, we finally obtain an improvement on the Hasse-Weil bound for Artin-Schreier curves.
This paper is organized as follows. In Section \ref{preliminary}, we introduce the basic results and theory on algebraic function fields and coding theory, such as Hilbert's ramification theory, conductors, Artin symbols, cyclotomic function fields, ray class fields and linear codes. In Section \ref{sec: 3}, we determine the conductors of Artin-Schreier curves and cyclotomic function fields, and show that the function field of an Artin-Schreier curve is a subfield of some cyclotomic function field. In Section \ref{sec: 4}, we show that the upper bound of the number of rational places of Artin-Schreier curves is a function of the minimum distance of some linear code. Furthermore, this upper bound is better than the Hasse-Weil bound for Artin-Schreier curves. In Section \ref{sec: 5}, we determine the minimum distance of the above linear code and provide a tight upper bound for Artin-Schreier curves with $\deg f(x)=2$, and we provide some examples for Artin-Schreier curves for $\deg f(x)\ge 3$ with the help of the software Magma.
\section{Preliminaries}\label{preliminary}
In this section, we introduce the basic results and theory on algebraic function fields and coding theory, such as Hilbert's ramification theory, conductors, Artin symbols, cyclotomic function fields, ray class fields and linear codes. For more details, please refer to \cite{AT67, Au00, MX19, NX01,St09}.
\subsection{Hilbert's ramification theory}
Let $\F_q$ be the finite field with $q$ elements. Let $K$ be the rational function field $\F_q(x)$, where $x$ is a transcendental element over $\F_q$.
There are exactly $q+1$ rational places of $K$, that is, the finite place $P_{\Ga}$ corresponding to $x-\Ga$ for all $\Ga\in \F_q$ and the infinity place $\infty$ corresponding to $1/x$.
Let $F/\F_q$ be a function field with genus $g(F)$ over the full constant field $\F_q$, which is called a global function field. Let $\PP_F$ denote the set of places of $F$. The place of $F$ with degree one is called rational.
Let $E/\F_q$ be a finite extension of function fields $F/\F_q$. The Hurwitz genus formula yields
$$2{g(E)}-2=[E:F]\cdot (2g(F)-2)+\deg \text{ Diff}(E/F),$$
where $\text{Diff}(E/F)$ stands for the different of the extension $E/F$ (see \cite[Theorem 3.4.13]{St09}).
For a place $P\in \PP_F$ and a place $Q\in \PP_E$ with $Q|P$,
we denote by $d(Q|P), e(Q|P)$ the different exponent and ramification index of $Q|P$, respectively.
Then the different of $E/F$ is given by
$$\text{Diff}(E/F)=\sum_{Q\in\PP_E} d(Q|P) Q.$$
If $Q|P$ is unramified or tamely ramified, then $d(Q|P)=e(Q|P)-1$ by Dedekind's Different Theorem \cite[Theorem 3.5.1]{St09}. However, if
$Q|P$ is wildly ramified, that is, $e(Q|P)$ is divisible by $\text{char}(\mathbb{F}_{q})$, then it is more complicated to calculate the different exponent $d(Q|P)$. One way to find the different exponent $d(Q|P)$ is through high ramification groups and Hilbert's different formula.
The $i$-th ramification group $G_i(Q|P)$ of $Q|P$ for each $i\ge -1$ is defined by
$$G_i(Q|P)=\{ \sigma\in \Gal(E/F)| \nu_Q(\sigma(z)-z) \ge i+1 \text{ for all } z\in \mathcal{O}_Q\},$$
where $\mathcal{O}_Q$ stands for the integral ring of $Q$ in $E$ and $\nu_Q$ is the normalized discrete valuation of $E$ corresponding to the place $Q$.
If $Q|P$ is wildly ramified, then the different exponent $d(Q|P)$ is
$$d(Q|P)=\sum_{i=0}^{\infty} \Big{(} |G_i(Q|P)|-1\Big{)}$$
from Hilbert's Different Theorem \cite[Theorem 3.8.7]{St09}.
\subsection{Conductor}
Let $F/\F_q$ be a global function field and let $E/\F_q$ be a finite abelian extension of $F/\F_q$. Let $P\in \mathbb{P}_F$ be a unramified place in the extension $E/F$ and $Q$ be a place of $E$ lying over $P$. Let $a(Q|P)$ be the least non-negative integer $l$ such that the ramification groups $G_i(Q|P)$ are trivial for all $i\ge l$.
The conductor exponent $c_P(E/F)$ of $P$ in $E/F$ is defined by
\[c_P(E/F):=\frac{d(Q|P)+a(Q|P)}{e(Q|P)}.\]
The conductor of $E/F$ is an effective divisor of $F$ given by
\[\text{Cond}(E/F):=\sum_{P\in \mathbb{P}_F} c_P(E/F) P.\]
The following two lemmas are useful to determine conductor exponents of places in abelian extensions (see \cite[Theorem 2.3.4]{NX01} and \cite[Lemma 2.3.7]{NX01}).
\begin{lemma}\label{lem: 2.1}
Let $E/F$ be a finite abelian extension of global function fields and let $P$ be a place of $F$. Then
\begin{itemize}
\item[(i)] $P$ is unramified in $E/F$ if and only if $c_P(E/F)=0$.
\item[(ii)] $P$ is tamely ramified in $E/F$ if and only if $c_P(E/F)=1$.
\item[(iii)] $P$ is widely ramified in $E/F$ if and only if $c_P(E/F)\ge 2$.
\end{itemize}
\end{lemma}
\begin{lemma}\label{lem: 2.3}
Let $K\subseteq F\subseteq E$ be three global function fields with $E/K$ being a finite abelian extension. Let $P$ be a place of $K$ and $Q$ be a place of $F$ lying over $P$. Then we have
$$c_P(F/K)\le c_P(E/K)\le \max\{c_P(F/K),c_Q(E/F)\}.$$
\end{lemma}
\subsection{Artin symbol}
Let $E/\F_q$ be a finite Galois extension of $F/\F_q$. Let $P\in \mathbb{P}_F$ be a unramified place in $E/F$ and $Q$ be a place of $E$ lying over $P$. Let $Z$ be the decomposition field of $Q$ over $P$. There exists a unique automorphism $\sigma\in \Gal(E/Z)$ such that $$\sigma(z)\equiv z^{q^{\deg(P)}}(\text{mod } Q) \text{ for all } z\in \mathcal{O}_Q. $$
This unique automorphism $\sigma$ is called the Frobenius symbol of $Q$ over $P$ and denoted by $\left[\frac{E/F}{Q}\right]$.
Furthermore, if $E/F$ is abelian, then the Frobenius symbol $\left[\frac{E/F}{Q}\right]$ does not depend on the choice of $Q$, but only on the place of $P$.
Hence, the Frobenius symbol $\left[\frac{E/F}{Q}\right]$ can be written as $\left[\frac{E/F}{P}\right]$ and called the Artin symbol of $P$ in $E/F$.
The following result can be found in \cite[Proposition 1.4.12]{NX01}. It characterizes whether a place splits completely in an abelian extension in terms of Artin symbols.
\begin{lemma}\label{lem: 2.0}
Let $E/K$ be a finite abelian extension and let $F$ be a subfield of $E/K$. Suppose that a place $P\in \mathbb{P}_K$ is unramified in the extension $E/K$. Then $P$ splits completely in $F/K$ if and only if the Artin symbol $\left[\frac{E/K}{P}\right]$ belongs to $\Gal(E/F)$.
\end{lemma}
\subsection{Cyclotomic function fields}
In this subsection, we briefly review some of the fundamental notions and results of cyclotomic function fields. The theory of cyclotomic function fields was developed in the language of function fields by Hayes (see \cite{Ha74, NX01}).
Let $q$ be a prime power. Let $x$ be an indeterminate over $\F_q$, $R=\F_q[x]$ the polynomial ring, $K=\F_q(x)$ the quotient field of $R$, and $K^{ac}$ the algebraic closure of $K$. Let $\varphi$ be the endomorphism given by $$\varphi(z)=z^q+xz $$ for all $z\in K^{ac}$. Define a ring homomorphism
$$R\rightarrow \text{End}_{\mathbb{F}_q}(K^{ac}), f(x)\mapsto f(\varphi).$$
Then the $\F_q$-vector space of $K^{ac}$ is made into an $R$-module by introducing the following action of $R$ on $K^{ac}$, namely,
$$ z^{f(x)}=f(\varphi)(z)$$ for all $f(x)\in R$ and $z\in K^{ac}$. For a nonzero polynomial $M\in R$, we consider the set of $M$-torsion points of $K^{ac}$ defined by $$\Lambda_M=\{z\in K^{ac}| z^M=0\}.$$
In fact, $z^M$ is a separable polynomial of degree $q^d,$ where $d=\deg(M)$. The cyclotomic function field over $K$ with modulus $M$ is defined by the subfield of $K^{ac}$ generated over $K$ by all elements of $\Lambda_M$, and it is denoted by $K(\Lambda_M)$. In particular, we list the following facts:
\begin{proposition}
\label{genusofcyclotomic}
Let $P$ be a monic irreducible polynomial of degree $d$ in $R$ and let $n$ be a positive integer. Then
\begin{itemize}
\item[\rm (i)] $[K(\Lambda_{P^n}):K]=\phi(P^n)$, where $\phi(P^n)$ is the Euler function of $P^n$, i.e., $\phi(P^n)=q^{(n-1)d}(q^d-1)$.
\item[\rm (ii)] ${\rm Gal}(K(\Lambda_{P^n})/K) \cong (\mathbb{F}_q[x]/(P^{n}))^*.$ The Galois automorphism $\sigma_f$ associated to $\overline{f}\in (\mathbb{F}_q[x]/(P^{n}))^*$ is determined by $\sigma_f(\lambda)=\lambda^f$ for $\lambda\in \Lambda_{P^{n}}$.
\item[\rm (iii)]The zero place of $P$ in $K,$ also denoted by $P$, is totally ramified in $K(\Lambda_{P^n})$ with different exponent $d_P(K(\Lambda_{P^n})/K)=n(q^d-1)q^{d(n-1)}-q^{d(n-1)}$. All other finite places of $k$ are unramified in $K(\Lambda_{P^n})/K$.
\item[\rm (iv)]The infinite place $\infty$ of $K$ splits into $\phi(P^n)/(q-1)$ places of $K(\Lambda_{P^n})$ and the ramification index $e_{\infty}(K(\Lambda_{P^n})/K)$ is equal to $q-1$. In particular, $\mathbb{F}_q$ is the full constant field of $K(\Lambda_{P^n})$.
\item[\rm (v)] The genus of $K(\Lambda_{P^n})$ is given by $$2g(K(\Lambda_{P^n}))-2=q^{d(n-1)}\Big{[}(qdn-dn-q)\frac{q^d-1}{q-1}-d\Big{]}.$$
\end{itemize}
\end{proposition}
\subsection{Ray class fields}
Let $F/\F_q$ be a global function field. Let $P$ be a place of $F$ and let $F_P$ be the completion of $F$ at $P$.
We still use $P$ to stand for the place of $F_P$ lying over $P$. We use $\mathcal{O}_{F_P}$ and $U_{F_P}$ for the valuation ring of $F_P$ and the group of units of $\mathcal{O}_{F_P}$, respectively.
For a place of $P$ of $F$, we consider the $n$-th unit group of $P$ in $F_P$ defined by \[U_{F_P}^{(n)}=\{x\in \mathcal{O}_{F_P}: \nu_P(x-1)\ge n\}\]
and denote by $U_{F_P}^{(0)}$ the unit group of $P$ in $F_P$.
Let $J_F$ be the set of ideles of $F$ and let $C_F$ be the idele class group $J_F/F^*$ of $F$.
Let $S$ be a non-empty finite subset of $\mathbb{P}_F$ and let $D$ be an effective divisor of $F$.
Define the $S$-congruence subgroup mod $D$ as $J_S^D=\prod_{P\in S}F_P^* \times \prod_{P\notin S} U_{F_P}^{(m_P)}$. Its class group is defined by
$$C_S^D=(F^*\cdot J_S^D)/F^*.$$
Let $\mathcal{O}_S$ be the holomorphy ring of all functions in $F$ with poles in $S$.
From the weak approximation \cite[Theorem 1.3.1]{St09} or \cite[Proposition 2.4.3]{NX01}, the $S$-ray class group mod $D$ is isomorphic to $S$-ideal class group mod $D$, i.e.,
$$ C_F/C_S^D\cong J_F/(F^*\cdot J_S^D)\cong \text{Cl}_D(\mathcal{O}_S).$$
Note that $\text{Cl}_D(\mathcal{O}_S)$ is defined as the quotient of the group of fractional ideals of $\mathcal{O}_S$ prime to $D$ by its subgroup of principal ideals.
It follows that $C_S^D$ is a subgroup of $C_F$ with finite index. Then there exists a unique extension $F_S^D/F$ such that $(F^*\cdot N_{E/F}(J_E))/F^*=C_S^D$ from the existence theorem \cite[Theorem 2.5.1]{NX01}. The field $F_S^D$ is called the $S$-ray class field with modulus $D$.
In particular, if $S$ consists of only one place, an explicit construction of $F_S^D$ via rank one Drinfeld modules has been given by Hayes \cite{Ha79}.
Moreover, the degree of the extension $F_{S}^D/F$ can be found from \cite{Au00}.
\begin{lemma}\label{prop: 2.3}
Let $S$ be a set consisting of a unique place of $F/\F_q$ with degree $t>0$. Let $D=\sum_{j=1}^{s}c_jQ_j$ be a positive divisor of $E$ with $\mbox{supp}(D)\cap S=\emptyset$. Let $h_F$ be the class number of $F$. Then we have
$$[F_S^D:F]=\frac1{q-1}\cdot h_F\cdot t \cdot \phi(D),$$
where $\phi(D)$ is the Euler function of $D$, i.e., $\phi(D)=\prod_{j=1}^s(q^{\deg (Q_j)}-1)q^{(c_j-1)\deg (Q_j)}$.
\end{lemma}
The following lemma is useful to determine the relationship of abelian extensions and ray class fields (see \cite[Theorem 2.5.4]{NX01}).
\begin{lemma}\label{lem: 2.2}{\bf (Conductor Theorem)}
Let $E/F$ be a finite abelian extension of global function fields and let $S$ be a non-empty finite subset of $\mathbb{P}_F$ such that any place in $S$ splits completely in $E/F$. Denote the conductor Cond$(E/F)$ by $C$. Then
\begin{itemize}
\item[(i)] $E$ is a subfield of $F_S^C$.
\item[(ii)] If $D$ is a positive divisor of $F$ with supp$(D)\cap S=\emptyset$ and $E\subseteq F_S^D$, then $D\ge C$.
\end{itemize}
\end{lemma}
From the above lemma, it is easy to see that $F_S^D$ is the largest abelian extension $E$ over $F$ such that Cond$(E/F)\le D$ and every place in $S$ splits completely in $E$.
\subsection{Linear codes}
In this subsection, we briefly discuss linear codes. The reader may refer to \cite{LX04} for the details.
A linear code $C$ of length $n$ over a finite field $\F_q$ is an $\F_q$-subspace of $\F_q^n$. The dimension of $C$, denoted by $\dim_{\F_q}(C)$, is defined to be the $\F_q$-dimension of $C$ as a vector space over $\F_q$. An element of $C$ is called a codeword. The Hamming weight of a vector in $\F_q^n$ is defined to be the number of nonzero coordinates. If $\dim_{\F_q}(C)>0$, then minimum distance of $C$ is defined to be the smallest Hamming weight of nonzero codewords in $C$.
An $m\times n$ matrix $G$ with entries in $\F_q$ is called a generator matrix of $C$ if (i) every row of $G$ is a codeword; and (ii) every codeword is a linear combination of rows of $G$. Note that most of textbooks require that the rows of $G$ are linearly independent. However, for convenience, we do not require this condition in this paper.
The following result that characterizes the minimum distance of $C$ in terms of $G$ is well known in the coding community. However, this result is not explicitly stated in literatures. For completeness, we provide a proof below.
\begin{lemma}\label{lem:2.3} Let $G$ be a generator matrix of $C$ of dimension $k$. Then the minimum distance of $C$ is $d$ if and only if
\begin{itemize}
\item[{\rm (i)}] any $n-d+1$ columns of $G$ form a matrix of rank $k$; and
\item[{\rm (ii)}] there exist $n-d$ columns of $G$ that form a matrix of rank at most $k-1$.
\end{itemize}
\end{lemma}
\begin{proof} Let $H$ be a $k\times n$ submatrix of $G$ such that the rows of $H$ form a basis of $C$.
Assume that the minimum distance of $C$ is $d$. Then there exists a codeword $\bc$ of $C$ with Hamming weight $d$. Without loss of generality, we may assume that the last $d$ positions of $\bc$ are nonzero. As $\bc$ is a nonzero linear combination of the rows of $H$, this implies that the first $n-d$ rows are linearly independent, i.e., the rank of the first $n-d$ rows of $H$ is less than $k$. As every row of $G$ is a linear combination of the rows of $H$, the rank of the first $n-d$ rows of $H$ is equal to the rank of the first $n-d$ rows of $G$. This proves (ii). Now suppose that, without loss of generality, the first $n-d+1$ columns of $G$ form a matrix of rank less than $k$, then the first $n-d+1$ columns of $G$ form a matrix of rank less than $k$ as well. This implies that there exists a nonzero codeword with the first $n-d+1$ positions equal to $0$, i.e., this codeword has Hamming weight at most $d-1$. This contradicts the fact that $d(C)=d$.
Now assume that both (i) and (ii) hold. Again without loss of generality, we assume that the first $n-d$ columns of $G$ has rank less than $k$. Then the fist $n-d$ columns of $G$ has rank less than $k$ as well. There there is a nonzero codeword $\bc$ that is a linear combination of rows of $H$ such that the first $n-d$ positions are zero. This implies that the Hamming weight of $\bc$ is at most $d$, i.e., $d(C)\le d$. Suppose that $d(C)<d$, then in a similar way one can show that there are $n-k+1$ columns that form a matrix of rank less than $k$. This conviction shows that $d(C)\ge d$. The proof is completed.
\end{proof}
\section{Artin-Schreier curves as subfields of cyclotomic function fields}\label{sec: 3}
In this section, we will determine the conductors of Artin-Schreier curves and the cyclotomic function fields over the rational function field $\F_q(x)$ with modulus $x^{-n-1}$ for any integer $n\ge 1$. In particular, we will show that the function field of an Artin-Schreier curve $y^p-y=f(x)$ with $f(x)\in \F_q[x]$ can be viewed as a subfield of some cyclotomic function field. From now onwards, we denote by $K$ the rational function field $\F_q(x)$.
\subsection{Artin-Schreier curves}
Let $p$ be a prime and let $q=p^m$ for some integer $m\ge 1$. Consider the Artin-Scherier curve defined by \[y^p-y=f(x),\]
where $f(x)$ is a polynomial of degree $r$ in $\F_q[x]$.
The corresponding function field is given by $E_f:=\F_q(x,y)$ with $y^p-y=f(x).$
From \cite[Proposition 3.7.8]{St09}, the properties of Artin-Schreier curves can be summarized as follows.
\begin{lemma}\label{lem: 3.1}
Let $K/\F_q$ be the rational function field of characteristic $p>0$. Let $f(x)$ be a polynomial in $\F_q[x]$ of degree $r$ with $\gcd(r,p)=1$.
Let \[E_f=K(y)=\F_q(x,y) \text{ with } y^p-y=f(x).\] For any place $P\in \mathbb{P}_K$, we define the integer $m_P$ by
$$
m_P=\begin{cases}
\ell &\text{ if } \exists z\in K \text{ satisfying } \nu_P(u-(z^p-z))=-\ell<0 \text{ and } \ell \not \equiv 0 (\text{mod }p),\\
-1 & \text{ if } \nu_P(u-(z^p-z))\ge 0 \text{ for some } z\in K.
\end{cases}$$
Then we have:
\begin{itemize}
\item[(a)] $E_f/K$ is a cyclic Galois extension of degree $p$. The automorphisms of $E_f/K$ are given by $\sigma(y)=y+u$ with $u=0,1,\cdots,p-1.$
\item[(b)] $P$ is unramified in $E_f/K$ if and only if $m_P=-1$.
\item[(c)] $P$ is totally ramified in $E_f/K$ if and only if $m_P>0$. Denote by $P^\prime$ the unique place of $E_f$ lying over $P$. Then the different exponent $d(P^\prime|P)$ is given by \[d(P^\prime|P)=(p-1)(m_P+1).\]
\item[(d)] The infinity place $\infty$ of $K$, i.e., the pole of $x$, is the unique ramified place in $E_f/K$, the full constant field of $E_f$ is $\F_q$ and
\[g(E_f)=\frac{(p-1)(r-1)}{2}.\]
\end{itemize}
\end{lemma}
\begin{proposition}\label{prop: 3.2}
Let $f(x)$ be a polynomial in $\F_q[x]$ of degree $r\ge 1$ with $\gcd(r,p)=1$. Let $\infty$ be the pole of $x$ in the rational function field $K$. Denote by $E_f$ the Artin-Schreier curve defined by $y^p-y=f(x)$. Then the conductor of $E_f/K$ is \[ \text{Cond}(E_f/K)=(r+1) \infty.\]
\end{proposition}
\begin{proof}
Let $P_\infty$ be the place of $E_f$ lying above $\infty$.
From Lemma \ref{lem: 3.1}, $P_\infty|\infty$ is the unique ramified place in $E_f/K$.
Then we have $\nu_{\infty}(f(x))=-\deg(f(x))=-r<0 \text{ and } \gcd(r,p)=1$, i.e., $m_{\infty}=r$. Hence, the different exponent of $P_\infty|\infty$ is given by
$$d(P_\infty|\infty)=(p-1)(m_{\infty}+1)=(p-1)(r+1).$$
Let $\sigma$ be the automorphism of $E_f/K$ defined by $\sigma(y)=y+u$ with $u\in \F_p$ and let $t$ be a prime element of $P_\infty$.
Then we have $\nu_{P_\infty}(\sigma(t)-t)=m_{\infty}+1=r+1 \text{ for } \sigma\neq 1$ from the proof of \cite[Proposition 3.7.8]{St09}. Hence, the higher ramification groups of $P_\infty|\infty$ are given by
$$G_0(P_\infty|\infty)=G_1(P_\infty|\infty)=\cdots=G_r(P_\infty|\infty)=\Gal(E_f/K) \text{ and } G_{r+1}(P_\infty|\infty)=\{1\}.$$
Thus, the conductor exponent of $\infty$ in $E_f/K$ is
$$c_\infty(E_f/K)=\frac{d(P_\infty|\infty)+a(P_\infty|\infty)}{e(P_\infty|\infty)}=\frac{(p-1)(r+1)+(r+1)}{p}=r+1.$$
This completes the proof from Lemma \ref{lem: 2.1}.
\end{proof}
\subsection{Subfields of cyclotomic function fields}
Let $T=(x-\a)^{-1}$ for $\a\in \F_q$ and $n$ be a positive integer. Then we have $K=\F_q(x)=\F_q(T)$. Let $K_n$ be the cyclotomic function field $K(\Lambda_{T^{n+1}})$ with modulus $T^{n+1}$ over $K$. In fact, the cyclotomic function field $K_n$ is an abelian extension over $K$ of degree $q^n(q-1)$.
Moreover, we have the following facts from \cite{Ha74, MXY16}:
\begin{lemma}\label{lem: 3.3}
Let $K_n = K(\Lambda_{T^{n+1}})$ denote the cyclotomic function field with modulus $T^{n+1}$ over the rational function field $K$. Then one has
\begin{itemize}
\item[\rm (i)] ${\rm Gal}(K_n/K) \cong (\mathbb{F}_q[T]/(T^{n+1}))^*.$ The Galois automorphism $\sigma_f$ associated to $\overline{f}\in (\mathbb{F}_q[T]/(T^{n+1}))^*$ is determined by $\sigma_f(\lambda)=\lambda^f$ for any generator $\lambda\in \Lambda_{T^{n+1}}$.
\item[\rm (ii)]The zero place of $T$ in $K,$ which is the pole $\infty$ of $x$ in $K$, is totally ramified in $K_n$. Let $Q_\infty$ denote the unique place of $K_n$ lying over $\infty$. The different exponent of $Q_\infty|\infty$ is $d(Q_\infty|\infty)=(n+1)q^n(q-1)-q^n$.
\item[\rm (iii)]The pole place of $T$ in $K$, which is the zero place $P_\a$ of $x-\a$ in $K$, splits into $q^n$ rational places with ramification index $q-1$. In particular, $\mathbb{F}_q$ is the full constant field of $K_n$.
\item[\rm (iv)] A monic irreducible polynomial $P\in \F_q[T]$ is unramified in $K_n/K$ if $P$ does not divide $T^{n+1}$.
\item[\rm (v)] For a monic irreducible polynomial $P\in \F_q[T]$ not dividing $T^{n+1}$, the Artin symbol $\left[\frac{K_n/K}{P}\right]$ satisfies
$$\left[\frac{K_n/K}{P}\right]: \lambda\mapsto \lambda^P$$ for any generator $\lambda$ of $\Lambda_{T^{n+1}}$.
\item[\rm (vi)] The automorphism group of $K_n$ over $\F_q$ is given by $$\Aut(K_n/\F_q):=\{\sigma: K_n\rightarrow K_n| \sigma \text{ is an } \F_q\text{-automorphism of } K_n\}=\Gal(K_n/K).$$
\end{itemize}
\end{lemma}
In the following, we need to determine the conductor of $K_n/K$.
\begin{proposition}\label{prop: 3.4}
Let $\infty$ be the pole place of $x$ and let $P_\a$ be the zero place of $x-\a$ in $K$. Then the conductor of $K_n/K$ is
$$\text{Cond}(K_n/K)=(n+1)\cdot \infty+\min\{1, q-2\}\cdot P_\a.$$
Furthermore, let $F$ be the subfield of $K_n$ fixed by $\F_q^*$, then the conductor of $F/K$ is
$$\text{Cond}(F/K)=(n+1)\cdot \infty.$$
\end{proposition}
\begin{proof}
Let $P_\a$ be the zero place of $x-\a$ in $K$. In fact, $P_\a$ is the pole place of $T$ in $K$.
Let $Q_\a$ be any place of $K_n$ lying above $P_\a$. Then we have $e(Q_\a|P_\a)=q-1$ from Lemma \ref{lem: 3.3}(iii).
If $q>2$, then $Q_\a|P_\a$ is tamely ramified and the conductor exponent of $P_\a$ in $K_n/K$ is $1$ from Lemma \ref{lem: 2.1}(ii); otherwise, $Q_\a|P_\a$ is unramified and the conductor exponent of $P_\a$ in $K_n/K$ is $0$ from Lemma \ref{lem: 2.1}(i).
Let $\infty$ be the pole place of $x$. From Lemma \ref{lem: 3.3}(ii), $\infty$ is totally ramified in $K_n/K$.
Let $Q_\infty$ denote the unique place of $K_n$ lying over $\infty$. Then we have $e(Q_\infty|\infty)=q^{n}(q-1)$ and the following claim holds true.
{\bf Claim:} The conductor exponent of $\infty$ in $K_n/K$ is $n+1$.
For any automorphism $\sigma\in \Gal(K_n/K)$, there exists an equivalence class of a polynomial $f(T)=\sum_{i=0}^{n} a_iT^i$ in $\F_q[T]/(T^{n+1})$ such that $\sigma(\lambda)=\lambda^f$. Since $\lambda^T=\lambda^q+T\lambda$ and $\nu_{Q_\infty}(T)=q^n(q-1)$, we have
\begin{eqnarray*}
\nu_{Q_\infty}(\sigma(\lambda)-\lambda) &= &\nu_{Q_\infty}(\lam^f-\lam) \\&=& \nu_{Q_\infty}(a_{n}\lam^{T^{n}}+a_{n-1}\lam^{T^{n-1}}+\cdots+a_1\lam^T+(a_0-1)\lam)\\ &=& \begin{cases} 1, & \text{ if } a_0\neq 1\\ q, & \text{ if } a_1\neq 0, a_0=1\\ \cdots\\ q^{n}, & \text{ if } a_{n}\neq 0, a_{n-1}=\cdots=a_1=0, a_0=1. \end{cases}
\end{eqnarray*}
Hence, the orders of higher ramification groups $g_i=|G_i(Q_\infty|\infty)|$ can be determined explicitly as follows:
$g_0=q^{n}(n-1)$, $g_1=g_2=\cdots=g_{q-1}=q^{n}$, $g_q=\cdots=g_{q^2-1}=q^{n-1}, \cdots, g_{q^{n-1}}=\cdots=g_{q^{n}-1}=q$ and $g_{q^{n}}=1$.
Hence, the conductor of $\infty$ in $K_n/K$ is
\[ c_\infty(K_n/K)=\frac{d(Q_\infty|\infty)+a(Q_\infty|\infty)} {e(Q_\infty|\infty)}=\frac{(n+1)q^{n}(q-1)-q^{n}+q^{n}}{q^{n}(q-1)}=n+1. \]
Thus, the first part follows from Lemma \ref{lem: 3.3}(iv) and Lemma \ref{lem: 2.1}.
Let $F$ be the subfield of $K_n$ fixed by $\F_q^*$. From Galois theory, $K_n/F$ is a finite extension of degree $q-1$ and $[F:K]=q^n$. It follows that $P_\a$ is tamely ramified in $K_n/F$ with ramification index $q-1$ and $P_\a$ is unramified in $F/K$. From Lemma \ref{lem: 2.1}, the conductor exponent of $P_\a$ in $F/K$ is $c_{P_\a}(F/K)=0$.
Let $P_{\infty}$ be the restriction of $Q_{\infty}$ in $F$.
The place $\infty$ is totally ramified in $K_n/K$ with ramification index $(q-1)q^n$ and $P_{\infty}$ is tamely ramified in $K_n/F$. From Lemma \ref{lem: 2.1}, we have $c_{P_\infty}(K_n/F)=1$. Furthermore, from Lemma \ref{lem: 2.3}, we have $$c_\infty(F/K)\le c_{\infty}(K_n/K)=n+1\le \max\{ c_\infty(F/K), c_{P_\infty}(K_n/F)\}.$$
It follows that $c_\infty(F/K)=n+1$. Hence, we have $\text{Cond}(F/K)=(n+1)\cdot \infty.$
\end{proof}
Now we can obtain the main result of this section.
\begin{theorem}\label{thm: 3.5}
Let $f(x)$ be a polynomial in $\F_q[x]$ of degree $r$. Let $E_f=\F_q(x,y)$ be the function field of an Artin-Schreier curve defined by $y^p-y=f(x)$.
Assume that there are at least one rational place $P_\a \in \mathbb{P}_K$ such that $P_\a$ splits completely in $E_f/K$.
Then the Artin-Schreier function field $E_f$ is a subfield of the cyclotomic function field $K_{t}$ for any integer $t\ge r$.
\end{theorem}
\begin{proof}
From Lemma \ref{prop: 2.3}, we have $[K_{P_\a}^{(r+1)\infty}:K]=q^r.$ From Proposition \ref{prop: 3.4}, the place $P_\a$ is splitting completely in $F/K$ and $\text{Cond}(F/K)=(n+1)\cdot \infty.$ From Lemma \ref{lem: 2.2}, we have $F\subseteq K_{P_\a}^{(r+1)\infty}$. Together with the fact $[F:K]=q^r$, we have $K_{P_\a}^{(r+1)\infty}=F,$ i.e, the ray class field $K_{P_\a}^{(r+1)\infty}$ is the subfield $F$ of $K_r$ fixed by $\F_q^*$.
As we know $P_\Ga$ splits completely in $E_f/K$ and Cond$(E_f/K)=(r+1)\infty$ from Proposition \ref{prop: 3.2}, it follows that $E_f$ is a subfield of $K_{P_\a}^{(r+1)\infty}=F$ from Lemma \ref{lem: 2.2}. Hence $E_f$ is a subfield of $K_r$. Since $K_r$ is a subfield of $K_t$ for any $t\ge r$ from the definition of cyclotomic function fields, $E_f$ is a subfield of $K_t$ as well.
\end{proof}
From Lemma \ref{lem: 3.3}, the cyclotomic function field $K_r$ is an abelian extension of $K$ with Galois group ${\rm Gal}(K_r/K) \cong (\mathbb{F}_q[T]/(T^{r+1}))^*.$
It follows that $K_r/E_f$ is a Galois extension with Galois group $\Gal(K_r/E_f)$ being a subgroup of $\Gal(K_r/K)$.
From Galois theory, the Artin-Schreier function field $E_f$ is the subfield of $K_r$ fixed by $\Gal(K_r/E_f)$.
Hence, we can estimate the number of rational places of Artin-Schreier curves by studying fixed subfields of subgroups of $\Gal(K_r/K)\cong \Aut(K_r/\F_q)$ with index $p$, which is very similar to the method of systematically constructing maximal function fields via automorphism groups of Hermitian function fields \cite{BMXY13,GSX00,MX19}.
\section{An upper bound of the number of rational places of Artin-Schreier curves}\label{sec: 4}
Let $q=p^m$ be a prime power and let $\F_q$ be the finite field with $q$ elements, i.e., $\F_q=\{0,\a_1,\a_2,\cdots,\a_{q-1}\}$.
Let $\mathcal{G}=\Gal(K_r/K)/\F_q^* \cong (\mathbb{F}_q[T]/(T^{r+1}))^*/\F_q^*$.
The set of $\mathcal{G}$ consists of elements: $$\mathcal{G} \cong (\mathbb{F}_q[T]/(T^{r+1}))^*/\F_q^*=\{\F_q^*(1+a_0T+\cdots+a_rT^r+(T^{r+1})): a_i\in \F_q \text{ for } 1\le i\le r\}.$$
We can identify $1+a_0T+\cdots+a_rT^r$ with the equivalence class $\F_q^*(1+a_0T+\cdots+a_rT^r+(T^{r+1}))$.
For simplicity, we assume that $r<p$. The order of any non-identity element in $(\mathbb{F}_q[T]/(T^{r+1}))^*/\F_q^*$ is $p$, since
$$(1+a_1T+a_2T^2+\cdots+a_rT^r)^p=1+a_1^pT^p+a_2^pT^{2p}+\cdots+a_r^pT^{rp}\equiv 1 (\text{mod } T^{r+1}).$$
Hence, $\mathcal{G}$ can be viewed as a vector space over $\F_p$, i.e., $\mathcal{G} \cong (\mathbb{F}_q[T]/(T^{r+1}))^*/\F_q^*\cong \F_p^{rm}$.
In order to estimate the number of rational places of the Artin-Scherier function field $E_f$, we need to study the number of rational places $P$ of $K$ such that its Artin symbol $[\frac{K_r/K}{P}]$ is contained in $\Gal(K_r/E_f)$.
Let $G$ be any subgroup of $\Gal(K_r/K) \cong (\mathbb{F}_q[T]/(T^{r+1}))^*$ with index $[\Gal(K_r/K): G]=p$.
Now we consider the rational places of $K$ corresponding to linear polynomials $1+\b T$ for $\b\in \F_q^*$, which are equivalent to rational places with respect to $T+\b^{-1}$ in $\F_q(T)$ or $x-(\Ga-\Gb)$ in $\F_q(x)$. We need to estimate the size of the set $\{\b\in \F_q^*: 1+\b T\in G\}$.
We view $1+\a_iT$ as a column vector of dimension $rm$ in $\mathcal{G}$. Let $A=(1+\a_1T, 1+\a_2T, \cdots, 1+\a_{q-1}T)$ be the matrix generated by column vectors $1+\a_iT$ for $1\le i\le q-1$. We need to find the smallest $d$ such that any $n-d+1$ columns of $A$ form a submatrix of rank $rm$.
Assume that $\{1+\a_iT\}_{i\in I}$ with $I\subseteq [q-1]:=\{1,2,\cdots,q-1\}$ form a submatrix of rank $rm$. Then we have the following result.
\begin{lemma}\label{lem: 4.1}
The space spanned by
$\{1+\a_iT\}_{i\in I}$ with $I=\{i_1,i_2,\cdots,i_t\}\subseteq [q-1]$ has $\F_p$-dimension $rm$ if and only if $G_I:=\left(\begin{array}{cccc}\a_{i_1} & \a_{i_2} & \cdots &\a_{i_t}\\ \a_{i_1}^2 & \a_{i_2}^2 & \cdots &\a_{i_t}^2\\ \vdots & \vdots & \ddots & \vdots \\ \a_{i_1}^r & \a_{i_2}^r & \cdots &\a_{i_t}^r\end{array}\right)\in \F_p^{rm\times (q-1)}$ has the full rank $rm$,
here $\Ga_i^k$ is viewed as a column vector under a basis of $\F_q$ over $\F_p$ for $1\le k\le r$ and $i\in I$.
\end{lemma}
\begin{proof}
Assume that $\{1+\a_iT\}_{i\in I}$ with $I=\{i_1,i_2,\cdots,i_t\}\subseteq [q-1]$ generates the vector space $\mathcal{G}$ over $\F_p$.
For any $1+\sum_{k=1}^r b_kT^k+(T^{r+1})\in \mathcal{G}$, there exist elements $u_i$ for $i\in I$ such that
$$1+\sum_{k=1}^r b_kT^k\equiv \prod_{i\in I}(1+\a_iT)^{u_i} (\text{mod } T^{r+1}),$$
i.e., $\prod_{i\in I}(1+\a_iT)^{u_i} =1+\sum_{k=1}^r b_kT^k+T^{r+1}h(T)$ for some $h(T)\in \F_q[T]$.
Taking the logarithm and using the Taylor expansion of log function, we have
\begin{eqnarray*}
\ln \left(1+\sum_{k=1}^r b_kT^k+T^{r+1}h(T)\right)&=&\sum_{i\in I} u_i\ln (1+\a_iT)=\sum_{i\in I} u_i \sum_{j=1}^{\infty} \frac{(-1)^{j-1}\a_i^j}{j}T^j\\ &=&\sum_{j=1}^\infty \frac{(-1)^{j-1}}{j}\left(\sum_{i\in I}u_i\a_i^j\right)T^j.
\end{eqnarray*}
Comparing with the coefficients of $T^j$ at the both sides for $1\le j\le r$, we have
\begin{eqnarray*} \frac{(-1)^{j-1}}{j}\left(\sum_{i\in I}u_i\a_i^j\right)&=&\text{ the coefficient of } T^j \text{ in } \sum_{\ell=1}^n\frac{(-1)^{\ell-1}(b_1T+\cdots+b_rT^r)^\ell}{\ell}\\
&=& \text{ the coefficient of } T^j \text{ in } \sum_{\ell=1}^j\frac{(-1)^{\ell-1}(b_1T+\cdots+b_jT^j)^\ell}{\ell}.\end{eqnarray*}
Hence, we have $$\sum_{i\in I}u_i\a_i^j=(-1)^{j-1}jb_j+f_j(b_1,b_2,\cdots,b_{j-1}), $$
for some $f_j\in \F_q[X_1,X_2,\cdots,X_{j-1}]$. That is to say, the system of linear equations
$$\left(\begin{array}{cccc}\a_{i_1}& \a_{i_2}& \cdots & \a_{i_t} \\ \a_{i_1}^2 & \a_{i_2}^2 & \cdots & \a_{i_t} ^2 \\ \vdots & \vdots & \cdots & \vdots \\ \a_{i_1}^r & \a_{i_2}^r & \cdots & \a_{i_t} ^r\end{array}\right)
\left(\begin{array}{c}x_{i_1}\\ x_{i_2}\\ \vdots \\ x_{i_t}\end{array}\right)=\left(\begin{array}{c}b_{1}\\b_1^2-2b_2\\ \vdots \\ (-1)^{r-1}rb_r+f_r(b_1,b_2,\cdots,b_{r-1})\end{array}\right):={\bf b}$$
has at least one solution $(u_{i_1},u_{i_2},\cdots, u_{i_t})\in \F_p^t$ for every tuple $(b_1,b_2,\cdots,b_r)\in \F_q^r$.
We claim that the rank of $G_I$ must be $rm$. Suppose that the rank of $G_I$ is strictly less than $rm$. When $b_i$ runs through $\F_q^*$ for each $1\le i\le r$, there must exist an tuple $(b_1,b_2,\cdots,b_r)$ such that the rank of $G_I$ is strictly less than the augmented matrix $(G_I, {\bf b})$ in the above system of linear equations. Thus, we obtain a contradiction.
On the other hand, if $G_I$ has rank $rm$, then the above system of linear equations has at least a solution for each tuple $(b_1,b_2,\cdots,b_r)\in \F_q^r$. That is to say any element $1+\sum_{k=1}^r b_kT^k\in \mathcal{G}$ can be generated by $\{1+\a_iT\}_{i\in I}$ with $I=\{i_1,i_2,\cdots,i_t\}\subseteq [q-1]$.
\end{proof}
\begin{remark}
The function $f_j$ in the above proof can be recursively computed. For instance, for $1\le j\le 5$, it is easy to compute that
$$\begin{cases} \sum_{i\in I} u_i \a_i=b_1,\\ \sum_{i\in I} u_i \a_i^2=b_1^2-2b_2,\\ \sum_{i\in I} u_i \a_i^3=b_1^3-3b_1b_2+3b_3,\\ \sum_{i\in I} u_i \a_i^4=b_1^4-4b_1^2b_2+2b_2^2+4b_1b_3-4b_4,\\
\sum_{i\in I} u_i \a_i^5=b_1^5-5b_1^3b_2+5b_1^2b_3+5b_1b_2^2-5b_1b_4-5b_2b_3+5b_5.\\ \end{cases}$$
\end{remark}
Let $C$ be the code generated by
\begin{equation}\label{eq:xx0}A=\left(\begin{array}{cccc}\a_1& \a_2& \cdots & \a_{q-1} \\ \a_1^2 & \a_2^2 & \cdots & \a_{q-1}^2 \\ \vdots & \vdots & \cdots & \vdots \\ \a_1^r & \a_2^r & \cdots & \a_{q-1}^r\end{array}\right)\in \F_p^{rm\times (q-1)},\end{equation}
where $\a_j^i\in \F_q^*$ is viewed as a column vector of dimension $m$ under a fixed $\F_p$-isomorphism between $\F_q$ and $\F_p^m$ for each $1\le i\le r$ and $1\le j\le q-1$. For given $p^m$ and $r$, the code $C$ is uniquely determined up to equivalence. Hence, the minimum distance $d(C)$ of $C$ is a uniquely determined function of the variables $p, m$ and $r$. We denote by $d(p,m,r)$ the minimum distance $d(C)$ of $C$.
\begin{theorem}\label{thm: 4.3}
Let $f(x)\in\F_q[x]$ be a polynomial of degree at most $r$, then the number of rational places of Artin-Scherier function field $E_f=\F_q(x,y)$ defined by $y^p-y=f(x)$ is upper bounded by \begin{equation}\label{eq:3.3} N_f\le 1+p(q-d(p,m,r)).\end{equation}
Moreover, there exists a polynomial $h(x)\in\F_q[x]$ of degree at most $r$ such that $N_h= 1+p(q-d(p,m,r)).$
\end{theorem}
\begin{proof}
Assume that there are at least one rational place $P_\a \in \mathbb{P}_K$ such that $P_\a$ splits completely in $E_f/K$.
From Theorem \ref{thm: 3.5}, $E_f$ is the subfield of $K_r$ fixed by some subgroup $G$ of $\Gal(K_r/K)$. As $[E_f:K]=p$, the index $[\Gal(K_r/K):G]$ is equal to $p$. A rational place corresponding to $T+\Gb^{-1}$ in $\F_q(T)$ or $P_{\Ga-\Gb}$ in $\F_q(x)$ splits completely in $E_f/K$, i.e., $y^p-y=f(\Ga-\Gb)$ has $p$ solutions for $\Gb\in \F_q^*$, if and only if $1+\Gb T$ belong to $G$. Suppose that there are exactly $t$ rational places of $K$ that split completely in $E_f$, i.e., $N_f=1+pt$, then there exist pairwise distinct elements $\{\Ga_1,\dots,\Ga_{t-1}\}$ of $\F_q^*$ such that the space spanned by $\{1+\Ga_1 T,\dots, 1+\Ga_{t-1} T\}$ has dimension at most $rm-1$. This implies that $t-1\le (q-1)-d(p,m,r)$ by Lemma \ref{lem:2.3}. This proves the inequality \eqref{eq:3.3}.
By Lemma \ref{lem:2.3} again, there exists a subset $I\subseteq[q-1]$ of size $|I|=q-1-d(p,m,r)$ such that the space spanned by $\{1+\Ga_i T\}_{i\in I}$ has dimension at most $rm-1$. Let $G$ be an $\F_p$-subspace of dimension $mr-1$ that contains $\{1+\Ga_i T\}_{i\in I}$ and let $L$ be the subfield of $F$ fixed by $G$, where $F$ is the subfield of $K_r$ fixed by $\F_q^*$. Then $L/K$ is a Galois extension of degree $p$. Thus, we must have that $L=\F_q(x,y)$ with $y^p-y=h(x)/u(x)$, where $h(x),u(x)\in\F_q[x]$ with $\gcd(h(x),u(x))=1$. As the pole of $x$ is the unique place that ramifies in $L/K$, we must have $u(x)=1$. The completes the proof.
\end{proof}
\begin{theorem}\label{thm: 4.4}
Let $d(p,m,r)$ be the minimum distance $d(C)$ of $C$ defined as above. Our bound \eqref{eq:3.3} given in Theorem \ref{thm: 4.3} is upper bounded by the Serre bound for the number of rational places of Artin-Schreier curves, i.e.,
$$ 1+p(q-d(p,m,r)) \le 1+q+\frac{(r-1)(p-1)}{2}\lfloor 2\sqrt{q}\rfloor$$
\end{theorem}
\begin{proof}
From Theorem \ref{thm: 4.3}, there exists a polynomial $h(x)\in\F_q[x]$ of degree at most $r$ such that $N_h= 1+p(q-d(p,m,r)).$
From the Serre bound, the number of rational places of Artin-Schreier curve $y^p-y=h(x)$ is upper bounded $$N_h\le 1+q+\frac{(\deg h(x)-1)(p-1)}{2}\lfloor 2\sqrt{q}\rfloor \le 1+q+\frac{(r-1)(p-1)}{2}\lfloor 2\sqrt{q}\rfloor.$$
This completes the proof.
\end{proof}
\begin{remark}
By making use of the Serre bound, we derive a lower bound on the minimum distance of the linear code generated by the matrix of \eqref{eq:xx0}, that is,
$$ d(p,m,r)\ge q-p^{m-1}-\frac{(r-1)(p-1)}{2p}\lfloor 2\sqrt{q}\rfloor.$$
If one can provide a good lower bound on the minimum distance $d(p,m,r)$, then the Serre bound for the Artin-Schreier curves defined by $y^p-y=f(x)$ can be improved.
\end{remark}
\iffalse
Finally in this section, by making use of the Serre bound, we derive a lower bound on the minimum distance of the linear code generated by the matrix of \eqref{eq:xx0}
\begin{theorem}\label{thm: 4.4}
Let $d(p,m,r)$ be the minimum distance $d(C)$ of $C$ of the linear code generated by the matrix of \eqref{eq:xx0}.
Then we have
$$ d(p,m,r)\ge q-p^{m-1}-\frac{(r-1)(p-1)}{2p}\lfloor 2\sqrt{q}\rfloor.$$
\end{theorem}
\begin{proof}
From Theorem \ref{thm: 4.3}, there exists a polynomial $h(x)\in\F_q[x]$ of degree at most $r$ such that $N_h= 1+p(q-d(p,m,r)).$
From the Serre bound, the number of rational places of Artin-Schreier curve $y^p-y=h(x)$ is upper bounded $$N_h\le 1+q+\frac{(\deg h(x)-1)(p-1)}{2}\lfloor 2\sqrt{q}\rfloor \le 1+q+\frac{(r-1)(p-1)}{2}\lfloor 2\sqrt{q}\rfloor.$$
Hence, we have $$ d(p,m,r)\ge q-p^{m-1}-\frac{(r-1)(p-1)}{2p}\lfloor 2\sqrt{q}\rfloor.$$
This completes the proof.
\end{proof}
In the following, we consider the Artin-Schreier curve $y^p-y=f(x)$ over $\F_p$ with $\deg f(x)=r<p$.
Let $C$ be the code generated by
$$A=\left(\begin{array}{cccc}\a_1& \a_2& \cdots & \a_{p-1} \\ \a_1^2 & \a_2^2 & \cdots & \a_{p-1}^2 \\ \vdots & \vdots & \cdots & \vdots \\ \a_1^r & \a_2^r & \cdots & \a_{p-1}^r\end{array}\right),$$
where $\a_i\in \F_p^*$ for each $1\le i\le r$.
\begin{corollary}\label{cor: 4.5}
Let $E_f/\F_p$ be the Artin-Schreier curve $y^p-y=f(x)$ over $\F_p$ with $\deg f(x)=r<p$ and let $C$ be the code generated by $A$ defined as above.
Then the number of rational places of $E_f$ is upper bounded by
$$N(E_f)\le 1+p(p-d(C))=1+pr.$$
\end{corollary}
\begin{proof}
From the Vandermonde matrix, any $r$ columns of $C$ are linearly independent and there are $r-1$ columns of $G$ of rank $r-1$.
Hence, the minimum distance of $C$ is $d(C)=n-r+1=p-r$ from Lemma \ref{lem:2.3}.
The code $C$ generated by the above matrix $A$ is a $p$-ary $[p-1,r,p-r]$ code. From Theorem \ref{thm: 4.3}, the number of rational places of $E_f$ is upper bounded by $N(E_f)\le 1+p(p-d(C))=1+pr.$
\end{proof}
\begin{remark}
The above Corollary \ref{cor: 4.5} can be proved from Lemma \ref{lem: 3.1} as well.
The Hasse-Weil bound of the number of rational places of Artin-Schreier curve $E_f/\F_p$ is $$1+p+2\frac{(r-1)(p-1)}{2}\sqrt{p}=1+p+(r-1)(p-1)\sqrt{p}.$$
Hence, the Hasse-Weil bound of Artin-Schreier curves can be improved for the case of prime finite field $\F_q=\F_p$, i.e., $m=1$. Roughly speaking, a factor $\sqrt{p}$ can be removed from the Hasse-Weil bound $N-p-1\le (p-1)(r-1)\sqrt{p}$.
\end{remark}
\fi
\section{Bounds and examples}\label{sec: 5}
In Section \ref{sec: 4}, we have shown that our bound \eqref{eq:3.3} given in Theorem \ref{thm: 4.3} is upper bounded by the Serre bound for the number of rational places of an Artin-Schreier curve $E_f$, i.e.,
$$N_f\le 1+p(q-d(p,m,r)) \le 1+q+\frac{(r-1)(p-1)}{2}\lfloor 2\sqrt{q}\rfloor.$$
In order to obtain a tight upper bound on the number of rational places of Artin-Schreier curve $E_f=\F_q(x,y)$ defined by $y^p-y=f(x)$, it remains to determine the minimum distance $d(p,m,r)$ of $C$ which is generated by the matrix
$$A=\left(\begin{array}{cccc}\a_1& \a_2& \cdots & \a_{q-1} \\ \a_1^2 & \a_2^2 & \cdots & \a_{q-1}^2 \\ \vdots & \vdots & \cdots & \vdots \\ \a_1^r & \a_2^r & \cdots & \a_{q-1}^r\end{array}\right)\in \F_p^{rm\times (q-1)},$$
where $\a_j^i\in \F_q^*$ is viewed as a column vector of dimension $m$ under a fixed $\F_p$-isomorphism between $\F_q$ and $\F_p^m$ for each $1\le i\le r$ and $1\le j\le q-1$.
However, the function of minimum distances $d(p,m,r)$ hasn't been determined explicitly.
In this section, we will determine the minimum distance $d(p,m,r)$ for $r=2$ and provide many examples of minimum distance $d(p,m,r)$ for $r\ge 3$ with the help of the software Magma.
\subsection{Bounds for $\deg f(x)=2$}
In this subsection, we will determine the minimum distance of the code $C$ defined by $$\left(\begin{array}{cccc}\a_{1} & \a_{2} & \cdots & \a_{q-1} \\ \a_{1}^2 & \a_{2}^2 & \cdots & \a_{q-1}^2 \end{array}\right),$$
where each $\a_j^i$ is viewed as a column vector under a fixed $\F_p$-basis $\{\r_1,\cdots,\r_m\}$ of $\F_q$ for $1\le i\le 2$ and $1\le j\le q-1$.
\begin{lemma}\label{lem: 5.1}
Let $C$ be a code defined by the above matrix. Then we have
\[C=\{(\text{Tr}(ax^2+bx))_{x\in \F_q^*}|\; a,b\in \F_q\}.\]
\end{lemma}
\begin{proof}
Let $\r_1,\r_2,\cdots,\r_m$ be a basis of $\F_q$ over $\F_p$. Then there exist a unique expression for $\a_i=\sum_{j=1}^m a_{ij}\r_j$ and $\a_i^2=\sum_{j=1}^m b_{ij}\r_j$ for some $a_{ij},b_{ij}\in \F_p$, respectively.
Let $G$ be the matrix defined by
$$G=\left(\begin{array}{cccc}a_{11} & a_{21} & \cdots & a_{q-1,1} \\ \vdots & \vdots & \cdots & \vdots \\ a_{1m} & a_{2m} &\cdots & a_{q-1,m}\\
b_{11} & b_{21} & \cdots & b_{q-1,1} \\ \vdots & \vdots & \cdots & \vdots \\ b_{1m} & b_{2m} &\cdots & b_{q-1,m}
\end{array}\right).$$
The code $C$ is generated by the rows of above matrix $G$.
Let $\{\b_1,\b_2,\cdots,\b_m\} $ be the dual basis of $\{\r_1,\r_2,\cdots,\r_m\}$. Then we have
$$(\text{Tr}(\b_i\a_1),\text{Tr}(\b_i\a_2),\cdots,\text{Tr}(\b_i \a_{q-1}))=(a_{1i},a_{2i},\cdots,a_{q-1,i})$$ and
$$(\text{Tr}(\b_i\a_1^2),\text{Tr}(\b_i\a_2^2),\cdots,\text{Tr}(\b_i \a_{q-1}^2))=(b_{1i},b_{2i},\cdots,b_{q-1,i}).$$
Hence, we have
\[C=\{(\text{Tr}(ax^2+bx))_{x\in \F_q^*}|\; a,b\in \F_q\}.\]
\end{proof}
In order to determine the minimum distance of $C$, it is sufficient to determine the size of the set $\{x\in \F_q^*: \text{Tr}(ax^2+bx)=0\}$.
Let $p$ be an odd prime, $q=p^ m$ and let $\zeta_1,\zeta_2,\cdots,\zeta_m$ be a basis of $\F_q$ over $\F_p$. Then there exists a unique linear combination $x=\sum_{i=1}^m x_i\zeta_i$ with $x_i\in \F_p$.
For any nonzero element $a\in \F_q^*$,
\begin{eqnarray*}
\text{Tr}(ax^2)=\text{Tr}(a( \sum_{i=1}^m x_i\zeta_i)^2)=\sum_{i,j=1}^m \text{Tr}(a\zeta_i\zeta_j)x_ix_j
\end{eqnarray*}
is a quadratic form in $n$ indeterminates $x_1,x_2,\cdots,x_m$ over $\F_p$. The function $\text{Tr}(ax^2)$ is non-degenerate quadratic form, since $a(x+z)^2-ax^2=2axz+z^2$ is permutation of $\F_q$ for any nonzero $z\in \F_q^*$.
The following two lemmas are very useful to determine the number of solutions of equations in quadratic forms from \cite[Theorem 6.26 and 6.27]{LN83}.
Let $v$ be an integer-valued function defined by $v(c)=-1$ for $c\in \F_q^*$ and $v(0)=q-1$.
\begin{lemma}\label{lem: 5.2}
Let $f$ be a nondegenerate quadratic form over $\F_q$ for odd prime power $q$ in an even number $n$ of indeterminates. Then for $c\in \F_q$ the number of the equation
$f(x_1,x_2,\cdots,x_n)=c$ in $\F_q^n$ is $$q^{n-1}+v(c)q^{(n-2)/2} \eta((-1)^{n/2}\Delta),$$
where $\eta$ is the quadratic character of $\F_q$ and $\Delta$ is the determinant of $f$.
\end{lemma}
\begin{lemma}\label{lem: 5.3}
Let $f$ be a nondegenerate quadratic form over $\F_q$ for odd prime power $q$ in an odd number $n$ of indeterminates. Then for $c\in \F_q$ the number of the equation
$f(x_1,x_2,\cdots,x_n)=c$ in $\F_q^n$ is $$q^{n-1}+q^{(n-1)/2} \eta((-1)^{(n-1)/2}c\Delta),$$
where $\eta$ is the quadratic character of $\F_q$ and $\Delta$ is the determinant of $f$.
\end{lemma}
\begin{proposition}\label{prop: 5.4}
Let $p$ be an odd prime, $q=p^ m$and $C$ be the code defined as above.
If $m$ is even, then the minimum distance of $C$ is $$d(p,m,2)=(p-1)p^{m-1}-(p-1)p^{\frac{m-2}{2}}.$$
If $m$ is odd, then the minimum distance of $C$ is $$d(p,m,2)=(p-1)p^{m-1}-p^{\frac{m-1}{2}}.$$
In particular, the code $C$ is a $q$-ary $[q-1,2m,d(p,m,2)]$ linear code.
\end{proposition}
\begin{proof}
As we have shown that $\text{Tr}(ax^2)$ is a nondegenerate quadratic form, the equation
\begin{eqnarray*}
\text{Tr}(ax^2+bx)=\text{Tr}\left(a( \sum_{i=1}^m x_i\zeta_i)^2+b\sum_{i=1}^m x_i\zeta_i\right)=\sum_{i,j=1}^m \text{Tr}(a\zeta_i\zeta_j)x_ix_j+\sum_{i=1}^m \text{Tr}(b\zeta_i)x_i=0
\end{eqnarray*}
is equivalent to the equation $a_1y_1^2+a_2y_2^2+\cdots+a_my_m^2+b_1y_1+b_2y_2+\cdots+b_my_m=0$ for $a_i\in \F_p^*$ and $b_i\in \F_p$ after a nonsingular linear transformation.
There is a one-to-one correspondence between $b\in \F_q$ and $(b_1,b_2,\cdots,b_m)\in \F_p^m$.
Let $z_i=y_i+b_i/2a_i$ for $1\le i\le m$ and $c_{a,b}=\sum_{i=1}^m b_i^2/4a_i$. Then the equation $\text{Tr}(ax^2+bx)=0$ is equivalent to the following form
$$a_1z_1^2+a_2z_2^2+\cdots+a_mz_m^2=c_{a,b}.$$
Let us first consider the case when $m$ is even.
From Lemma \ref{lem: 5.2}, the number of solutions of the equation with the non-degenerate quadratic form $\text{Tr}(ax^2+bx)=0$ is
$$p^{m-1}+v(c_{a,b}) p^{\frac{m-2}{2}} \eta((-1)^{\frac{m}{2}}\Delta_a), $$
where $\eta(\cdot)=(\frac{\cdot}{p})$ is the quadratic character or Legendre symbol of $\F_p$ and $\Delta_a=\prod_{i=1}^m a_i$ is the determinant of the quadratic form.
Hence, the possible smallest Hamming weight of nonzero codewords of $C$ is $q-p^{m-1}-(p-1)p^{\frac{m-2}{2}}.$
It remains to prove that there exists $a\in \F_q^*$ such that $\eta((-1)^{\frac{m}{2}}\Delta_a)=1$ for the quadratic form Tr$(ax^2)$.
Using the technique of double counting, we have
\begin{eqnarray*}
\sum_{a\in \F_q^*} |\{x\in \F_q: \text{Tr}(ax^2)=0\}|&=&\sum_{x\in \F_q} |\{a\in \F_q^*: \text{Tr}(ax^2)=0\}|\\&=&(q-1)(p^{m-1}-1)+(q-1)\\&=&p^{m-1}(q-1).
\end{eqnarray*}
Thus, there are exactly one-half elements $a\in \F_q^*$ such that $ \eta((-1)^{\frac{m}{2}}\Delta_a)=1$ and one-half elements $a\in \F_q^*$ such that $ \eta((-1)^{\frac{m}{2}}\Delta_a)=-1$. It follows that there exists an codeword in $C$ with Hamming weight $q-p^{m-1}-(p-1) p^{\frac{m-2}{2}}$.
Hence, the minimum distance of $C$ is $q-p^{m-1}-(p-1) p^{\frac{m-2}{2}}$.
If $m$ is odd, then the number of solutions of $\text{Tr}(ax^2+bx)=0$ is
$$N=p^{m-1}+p^{\frac{m-1}{2}} \eta((-1)^{\frac{m-1}{2}}c_{a,b}\Delta_a)$$
from Lemma \ref{lem: 5.3}. It is easy to check that $x^2+bx$ is a permutation of $\F_q$, we have
\begin{equation}\label{eq:8}
\sum_{b\in \F_q} |\{x\in \F_q: \text{Tr}(x^2+bx)=0\}|=p^{m-1}q.
\end{equation}
Moreover, there exists an element $b\in \F_q^*$ such that $c_{1,b}\neq 0$.
It follows that there are at least one element $b\in \F_q^*$ such that $ \eta((-1)^{\frac{m}{2}}c_{1,b}\Delta_1)=1$ from Equation \eqref{eq:8} and the Hamming weight of $\text{Tr}(x^2+bx)_{x\in \F_q^*}$ is $q-p^{m-1}- p^{\frac{m-1}{2}}$.
Hence, the minimum distance of $C$ is $q-p^{m-1}- p^{\frac{m-1}{2}}$.
\end{proof}
From Theorem \ref{thm: 4.3} and Proposition \ref{prop: 5.4}, we have the following tight bound for the number of rational places of Artin-Schreier curves defined by $y^p-y=f(x)$ with $\deg f(x)=2$.
\begin{theorem}\label{thm: 5.5}
The number of rational places of the function field $E_f=\F_q(x,y)$ of Artin-Scherier curve defined by $y^p-y=f(x)$ with $\deg f(x)=2$ is upper bounded by
$$N(E_f)\le 1+p(q-d(p,m,2))=\begin{cases} q+1+p^{\frac{m+1}{2}} & \text{ if } m \text{ is odd,}\\ q+1+(p-1)p^{\frac{m}{2}} & \text{ if } m \text{ is even.}\end{cases}$$
Moreover, the above upper bound can be achieved by the number of rational places of Artin-Scherier curves.
\end{theorem}
\begin{remark}
From Lemma \ref{lem: 3.1}, the genus of the Artin-Scherier curve $E_f$ is $$g(E_f)=\frac{(\deg f-1)(p-1)}{2}=\frac{p-1}{2}$$ and the Hasse-Weil upper bound is given by $$q+1+2g(E_f)\sqrt{q}=q+1+(p-1)\sqrt{q}.$$
If $m$ is even, the bound given in Theorem \ref{thm: 5.5} is the same as the Hasse-Weil bound. For even $m$, there are maximal curves given in the form of Artin-Schreier curves with genus $(p-1)/2$.
If $m$ is odd, the bound given in Theorem \ref{thm: 5.5} is better than the Serre bound. Roughly speaking, a factor $\sqrt{p}$ can be removed from the Hasse-Weil bound $N-q-1\le (p-1)\sqrt{q}$.
\end{remark}
\iffalse
\begin{example}
\begin{itemize}
\item[(1)] If $q=7^2$, then the code $C$ is a $[48,4,36]$ linear code. Then the number of rational places is upper bounded by $$N(E_f)\le 1+7\times (49-36)=92,$$
which is the same as the Hasse-Weil upper bound $q+1+2g(E_f)\sqrt{q}=49+1+2\times 3\times 7=92$.
\item[(2)] If $q=3^4$, then the code $C$ is a $[80,8,48]$ linear code. Then the number of rational places is upper bounded by $$N(E_f)\le 1+3\times (81-48)=100,$$
which is the same as the Hasse-Weil upper bound $q+1+2g(E_f)\sqrt{q}=81+1+2\times 1\times 9=100$.
\item[(3)] If $q=5^3$, then the code $C$ is a $[124,6,95]$ linear code. Then the number of rational places is upper bounded by $$N(E_f)\le 1+5\times (125-95)=151,$$
while the Serre bound $q+1+g(E_f)\cdot [2\sqrt{q}]=125+1+2\times [2\times \sqrt{125}]=170$. For more examples, please refer to Table \ref{table: 5.1}.
\item[(4)] If $q=3^5$, then the code $C$ is a $[242,10,153] $ linear code. Then the number of rational places is upper bounded by $$N(E_f)\le 1+3\times (243-153)=271,$$
while the Serre bound $q+1+g(E_f)\cdot [2\sqrt{q}]=243+1+1\times [2\times \sqrt{243}]=275$. For more examples, please refer to Table \ref{table: 5.2}.
\end{itemize}
\end{example}
\begin{table}[h]
\caption{$q=p^3, \deg f(x)=2$}
\label{table: 5.1}\vskip4pt
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline & $q=p^3$ & Parameters [n,k,d] & Serre bound & Our bound \\
\hline 1 & $5^3$ & [124,6,95] & 170 & 151 \\
\hline 2 & $7^3$ & [342,6,287] & 455 & 393 \\
\hline 3 & $11^3$ & [1330,6,1199] & 1692 & 1453 \\
\hline 4 & $13^3$ & [2196,6,2015] & 2756 & 2367\\
\hline 5 & $17^3$ & [4912,6,4607] & 6034 & 5203\\
\hline 6 & $19^3$ & [6858,6,6479] & 8345 & 7221 \\
\hline 7 & $23^3$ & [12166,6,11615] & 14588 & 12697 \\
\hline \hline
\end{tabular}
\end{table}
\begin{table}[h]
\caption{$q=p^5, \deg f(x)=2$}
\label{table: 5.2}\vskip4pt
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline & $q=p^5$ & Parameters [n,k,d] & Serre bound & Our bound \\
\hline 1 & $3^5$ & [242,10,153] & 275 & 271 \\
\hline 2 & $5^5$ & [3124,10,2475] & 3348 & 3251 \\
\hline \hline
\end{tabular}
\end{table}
\fi
\subsection{Examples for $\deg f(x)\ge 3$}
Unfortunately we can't determine the exact value of minimum distance $d(p,m,r)$ for $r\ge 3$.
However, we provide some examples of the minimum distance $d(p,m,r)$ with the help of the software Magma in this subsection.
\begin{example}
If $\deg f(x)=3$, then the genus of the Artin-Scherier curve $E_f$ is $$g(E_f)=\frac{(\deg f-1)(p-1)}{2}=p-1$$
and the Hasse-Weil upper bound is given by $$q+1+2g(E_f)\sqrt{q}=q+1+2(p-1)\sqrt{q}.$$
\begin{itemize}
\item[(1)] If $q=7^2$, then the code $C$ is a $ [48,6,34] $ linear code. From Theorem \ref{thm: 4.3}, the number of rational places is upper bounded by $$N(E_f)\le 1+7\times (49-34)=106,$$
while the Hasse-Weil bound is $q+1+2g(E_f)\cdot \sqrt{q}=49+1+2\times 6\times \sqrt{49}=134$. For more examples, please refer to Table \ref{table: 5.3}.
From the table, we can see that our bound is indeed better than the Hasse-Weil bound. Furthermore, if our bound achieves the Hasse-Weil bound, then there exist maximal function fields given by Artin-Schreier curves.
\item[(2)] If $q=5^3$, then the code $C$ is a $[124,9,90]$ linear code. From Theorem \ref{thm: 4.3}, the number of rational places is upper bounded by $$N(E_f)\le 1+5\times (125-90)=176,$$
while the Serre bound is $q+1+g(E_f)\cdot [2\sqrt{q}]=125+1+4\times [2\times \sqrt{125}]=214$. For more examples, please refer to Table \ref{table: 5.4}.
\end{itemize}
\end{example}
\begin{table}[h]
\caption{$q=p^2, \deg f(x)=3$}
\label{table: 5.3}\vskip4pt
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline & $q=p^2$ & Parameters [n,k,d] & Hasse-Weil bound & Our bound \\
\hline 1 & $5^2$ & [24,6,12] & 66 & 66 \\
\hline 2 & $7^2$ & [48,6,34] & 134 & 106 \\
\hline 3 & $11^2$ & [120,6,90] & 342 & 342 \\
\hline 4 & $13^2$ & [168,6,142] & 482 & 352\\
\hline 5 & $17^2$ & [288,6,240] & 834 & 834\\
\hline 6 & $19^2$ & [360,6,322] & 1046 & 742 \\
\hline 7 & $23^2$ & [528,6,462] & 1542 & 1542\\
\hline 8 & $29^2$ & [840,6,756] & 2466 & 2466\\
\hline 9 & $31^2$ & [960,6,898] & 2822 & 1954 \\
\hline 10 & $37^2$ & [1368,6,1294] & 4034 & 2776\\
\hline 11 & $41^2$ & [1680,6,1560] & 4962 & 4962 \\
\hline 12 & $43^2$ & [1848,6,1762] & 5462& 3742 \\
\hline 13 & $47^2$ & [2208,6,2070] & 6534 & 6534 \\
\hline 14 & $53^2$ & [2808,6,2652] & 8322 & 8322\\
\hline 15 & $59^2$ & [3480,6,3306] & 10326 & 10326 \\
\hline 16 & $61^2$ & [3720,6,3598] & 11042 & 7504 \\
\hline 17 & $67^2$ & [4488,6,4354] & 13334 & 9046 \\
\hline 18 & $71^2$ & [5040,6,4830] & 14982 & 14982\\
\hline 19 & $73^2$ & [5328,6,5182] & 14842 & 10732 \\
\hline 20 & $79^2$ & [6240,6,6082] & 18566 & 12562 \\
\hline 21& $83^2$ & [6888,6,6642] & 20502 & 20502\\
\hline \hline
\end{tabular}
\end{table}
\begin{table}[h]
\caption{$q=p^3, \deg f(x)=3$}
\label{table: 5.4}\vskip4pt
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline & $q=p^5$ & Parameters [n,k,d] & Serre bound & Our bound \\
\hline 1 & $5^3$ & [124,9,90] & 214 & 176 \\
\hline 2 & $7^3$ & [342,9,270] & 566 & 512 \\
\hline 3 & $11^3$ & [1330,9,1166] & 2052 & 1816 \\
\hline 4 & $13^3$ & [2196,9,1944] & 3314 & 3290 \\
\hline \hline
\end{tabular}
\end{table}
\begin{example}
If $\deg f(x)=4$, then the genus of $E_f$ is $$g(E_f)=\frac{(\deg f-1)(p-1)}{2}=\frac{3(p-1)}{2}, $$
and the Hasse-Weil upper bound is given by $$q+1+2g(E_f)\sqrt{q}=q+1+3(p-1)\sqrt{q}.$$
\begin{itemize}
\item[(1)] If $q=5^2$, then the code $C$ is a $ [24,8,12] $ linear code. From Theorem \ref{thm: 4.3}, the number of rational places is upper bounded by $$N(E_f)\le 1+5\times (25-12)=66,$$
while the Hasse-Weil bound is $q+1+2g(E_f) \sqrt{q}=25+1+2\times 6\times \sqrt{25}=86$. For more examples, please refer to Table \ref{table: 5.5}.
\item[(2)] If $q=5^3$, then the code $C$ is a $[124,12,83]$ linear code. From Theorem \ref{thm: 4.3}, the number of rational places is upper bounded by $$N(E_f)\le 1+5\times (125-83)=211,$$
while the Serre bound is $q+1+g(E_f)[2\sqrt{q}]=125+1+6\times [2\times \sqrt{125}]=258$.
\item[(3)] If $q=7^3$, then the code $C$ is a $[342,12,265]$ linear code. From Theorem \ref{thm: 4.3}, the number of rational places is upper bounded by $$N(E_f)\le 1+7\times (343-265)=547,$$
while the Serre bound is $q+1+g(E_f)[2\sqrt{q}]=343+1+9\times [2\times \sqrt{343}]=677$.
\end{itemize}
\end{example}
\begin{table}[h]
\caption{$q=p^2, \deg f(x)=4$}
\label{table: 5.5}\vskip4pt
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline $\deg f(x)=4$ & $q=p^2$ & Parameters [n,k,d] & Hasse-Weil bound & Our bound \\
\hline 1 & $5^2$ & [24,8,12] & 86 & 66 \\
\hline 2 & $7^2$ & [48,8,24] & 176 & 176 \\
\hline 3 & $11^2$ & [120,8,80] & 452 & 452 \\
\hline 4 & $13^2$ & [168,8,132] & 638 & 482\\
\hline 5 & $17^2$ & [288,8,240] & 1106 & 834\\
\hline 6 & $19^2$ & [360,8,288] & 1388 & 1388 \\
\hline 7 & $23^2$ & [528,8,440] & 2048 & 2048\\
\hline \hline
\end{tabular}
\end{table}
\begin{example}
If $\deg f(x)=5$, then the genus of $E_f$ is $$g(E_f)=\frac{(\deg f-1)(p-1)}{2}=2(p-1), $$
and the Hasse-Weil upper bound is given by $$q+1+2g(E_f)\sqrt{q}=q+1+4(p-1)\sqrt{q}.$$
If $q=7^2$, then the code $C$ is a $ [48,10,24] $ linear code. From Theorem \ref{thm: 4.3}, the number of rational places is upper bounded by $$N(E_f)\le 1+7\times (49-24)=176,$$
while the Hasse-Weil bound is $q+1+2g(E_f) \sqrt{q}=49+1+2\times 12\times \sqrt{49}=218$. For more examples, please refer to Table \ref{table: 5.6}.
\end{example}
\begin{table}[h]
\caption{$q=p^2, \deg f(x)=5$}
\label{table: 5.6}\vskip4pt
\begin{tabular}{||c|c|c|c|c|c|c||}
\hline
\hline $\deg f(x)=5$ & $q=p^2$ & Parameters [n,k,d] & Hasse-Weil bound & Our bound \\
\hline 1 & $7^2$ & [48,10,24] & 218 & 176 \\
\hline 2 & $11^2$ & [120,10,80] & 562 &452 \\
\hline 3 & $13^2$ & [168,10,132] & 794 & 482 \\
\hline \hline
\end{tabular}
\end{table}
From the above examples, we can see that our bound \eqref{eq:3.3} given in Theorem \ref{thm: 4.3} is better than the Hasse-Weil bound and Serre bound if the minimum distance $d(p,m,r)$ are explicitly determined.
\section{Conclusion}
In order to obtain a tight upper bound for the number of rational places of Artin-Schreier curves, it remains to determine the exact value or provide a good lower bound for the minimum distance of the code $C$ generated by the matrix
$$A=\left(\begin{array}{cccc}\a_1& \a_2& \cdots & \a_{q-1} \\ \a_1^2 & \a_2^2 & \cdots & \a_{q-1}^2 \\ \vdots & \vdots & \cdots & \vdots \\ \a_1^r & \a_2^r & \cdots & \a_{q-1}^r\end{array}\right)\in \F_p^{rm\times (q-1)},$$
where $\a_j^i\in \F_q^*$ are viewed as column vectors of dimension $m$ under a fixed basis of $\F_q$ over $\F_p$ for $1\le i\le r$ and $1\le j\le q-1$.
In this paper we determine the exact value of the minimum distance only for $r=2$. For $r\ge 3$, we provide some numerical examples.
In general, we leave it as an open research problem.
| 202,645
|
TITLE: How to compute isomorphism $V \simeq V^{**}$ (in Haskell)?
QUESTION [6 upvotes]: Since there is a canonical isomorphism between vector space $V$ and his dual dual space $V^{**}$, $\dim V \in \mathbb N \;$, I want to write it as a Haskell function.
This function is going to have a type of
$$ F : \left( \left(V \to \mathbb K \right) \to \mathbb K \right) \to V $$
Is it possible at all?
REPLY [2 votes]: I thought a bit more. Yes, one need to be able to point a basis in $V$.
But that doesn't imply a certain data structure --- one just need to provide an isomorphism $V \simeq \mathbb K^n \;$ (preferably as a class method, alongside with $+$ and $\cdot\;$). That seems to be enough to construct both $V^*$ and $V \simeq V^{**}\;$.
At first I wasn't sure $\xi : V \simeq \mathbb K^n\;$ will allow for $\eta : V^* \simeq \mathbb K^n\;$, such that $V^{**} \to V\;$ may go as $V^{**} \to \mathbb K^n \to V\;$. However an experiment reveals
$$\left((\eta_x(f), \; \eta_y(f)\right) = \left(f \xi^{-1}(1,0), \; f \xi^{-1}(0,1)\right)$$
$$\eta^{-1}(f_x, f_y) = v \mapsto \left(x \cdot \xi_x(v) + y \cdot \xi_y(v)\right)$$
to be the way to construct $V^* \simeq \mathbb K^n\;$.
A Haskell example for $\dim V = 2\;$ case:
{-# LANGUAGE TypeSynonymInstances, FlexibleInstances #-}
class Vector2D v where
(＋) :: v -> v -> v
(·) :: Double -> v -> v
toArithSpace :: v -> (Double, Double)
fromArithSpace :: (Double, Double) -> v
Casting $V^*\;$:
type Dual v = v -> Double
instance (Vector2D v) => Vector2D (Dual v) where
f＋g = \x -> f x + g x
(·) = \l f -> (\x -> l * f(x))
toArithSpace f = (f $ fromArithSpace (1,0), f $ fromArithSpace (0,1))
fromArithSpace (x, y) = \z -> x * (fst $ toArithSpace z) + y * (snd $ toArithSpace z)
and $\tau : V \simeq V^{**}\;$:
tauF :: Vector2D v => v -> (Dual (Dual v))
tauF x = \f -> f x
tauR :: Vector2D v => (Dual (Dual v)) -> v
tauR x = fromArithSpace (toArithSpace x)
To show that $\tau$ does not depend on $\xi : V \simeq \mathbb K^n \;$ lets's test two different $\xi$:
data Pair = Pair Double Double deriving Show
data Pair' = Pair' Double Double deriving Show
instance Vector2D Pair where
(Pair x1 y1)＋(Pair x2 y2) = Pair (x1+x1) (y1+y1)
l · (Pair x y) = Pair (l*x) (l*y)
toArithSpace (Pair x y) = (x+y,x-y)
fromArithSpace (x, y) = Pair (0.5*(x+y)) (0.5*(x-y))
instance Vector2D Pair' where
(Pair' x1 y1)＋(Pair' x2 y2) = Pair' (x1+x1) (y1+y1)
l · (Pair' x y) = Pair' (l*x) (l*y)
toArithSpace (Pair' x y) = (x+2*y,4*x-y)
fromArithSpace (x, y) = Pair' ((x+2*y)/9) ((4*x-y)/9)
*Main> (tauR.tauF) $ Pair 3 7
Pair 3.0 7.0
*Main> (tauR.tauF) $ Pair' 3 7
Pair' 3.0 7.0
Will that do?
| 218,818
|
\begin{document}
\maketitle
\begin{abstract}
Consider an analytic Hamiltonian system near its analytic invariant
torus $\mathcal T_0$ carrying zero frequency.
We assume that the Birkhoff normal form of the Hamiltonian at
$\mathcal T_0$ is convergent and
has a particular form: it is an analytic function of its
non-degenerate quadratic
part. We prove that in this case there is an analytic canonical
transformation---not just a formal power series---bringing
the Hamiltonian into its Birkhoff normal form.
\end{abstract}
\maketitle
\section{Introduction}
The goal of this paper is to study the convergence of
the transformations of an analytic Hamiltonian system in a
neighborhood of an invariant torus to the Birkhoff normal form.
Here we assume that
the frequency vector at the invariant torus is very resonant, hence already at the formal level, the existence of the
Birkhoff normal form has obstructions.
The main result, Theorem~\ref{main} below, will show that if the obstructions
for the formal equivalence between the system and its Birkhoff normal form vanish and the normal form is convergent and has
a particular form,
then the system is analytically equivalent to its normal form. Hence, this result
can be considered as a part of the rigidity program: identifying
obstructions for a weak form of equivalence whose vanishing
implies a stronger form of equivalence.
\subsection{Classical theory of normal forms: existence and uniqueness.}
Consider an analytic function
\beq\label{Ham1}
H(I,\theta)=\left< \lb_0,I\right> +\cO^2(I),
\eneq
where $\theta\in \T^d=\R^d/\Z^d$, $I\in (\R^d,0)$, $\left< \cdot,\cdot\right>$
denotes the usual scalar product in $\R^d$, and $\lambda_0\in \R^d$ is a constant
vector called the {\it frequency vector}. The Hamiltonian system
associated to it is
$
\dot I=\partial_{\theta} H(I,\theta), \
\dot \theta =- \partial_{I}H(I,\theta)
$.
Note that we are assuming the standard symplectic form. In particular, the set $\mathcal T_0:=\{0\}\times \T^d
$ is an invariant torus of this system.
We say that $H(I,\theta)$ {\it has a Birkhoff normal form (BNF)} $N(I)$ in
a neighborhood of $\mathcal T_0$ if $N(I)$ is a {\it formal} power series, and there
exists a {\it formal} symplectic transformation $\Psi(I,\theta)$,
tangent to the identity
$$
\Psi(I,\theta)=(I+\cO^2(I),\theta+\cO(I))
$$
such that
$$
H\circ \Phi(I,\theta)= N(I)
$$
in the sense of formal power series.
Any canonical coordinate change $\Phi(I,\theta)$ as above is called a {\it normalizing
transformation}. The following
fundamental result is called the Birkhoff normal form \cite{SiegelM71, MeyerHO}.
For $H(I,\theta)$ as above, assume that $\lambda_0$
satisfies a Diophantine condition: there
exist constants $(C,\tau)$ such that for all
$k\in \Z^d\setminus \{0\}$ we have
\beq\label{DC}
\left| \<\lb_0, k \> \right| \geq
C|k|^{-\tau}.
\eneq
Then $H(I,\theta)$ has a (formal) Birkhoff normal form. Moreover, if a
normal form exists and $\lambda_0$ is rationally independent, then
the Birkhoff normal form is unique (up to trivial changes relabelling the
actions). Note that the normalizing transformations are not unique, since
composing $\Phi(I,\theta)$ with any transformation that preserves $I$ gives a normalizing transformation.
Birkhoff normal form is an important tool in the study of Hamiltonian
systems. Already the assumption of existence and nondegeneracy of the normal
form has strong dynamical consequences (see, e.g., \cite{EFK15} Th.C).
The importance of the BNF becomes even stronger if the normal form is convergent, and even
more so if there exists an analytic normalizing transformation.
The standard way of constructing BNF, which we will review in more detail
later, is to proceed iteratively, devising transformations
that normalize $H(I,\theta)$ up to the coefficients of order $I^n$.
The normalization step involves solving
differential equations with analytic conditions. The
Diophantine conditions \eqref{DC} can be somewhat
weakened to subexponential growth ($ \lim_{N \to \infty}\frac{1}{N}
\log \sup_{|k| \le N } \left| \<\lb_0, k \> \right|^{-1} = 0$).
If $\lb_0$ is resonant, one cannot guarantee the existence of the
Birkhoff normal form even at the level of
formal power series, since
there may be some terms in the formal power series of $H$ that cannot
be eliminated by a canonical transformation. On the other hand,
there are, of course, systems (e.g the BNF itself, or changes of variables
from it) for which one can construct a BNF even in the resonant case.
Then one speaks of the
Birkhoff-Gustavson normal form \cite{Gustavson66}.
Analogous definitions and statements hold true for symplectic maps
in a neighborhood of a fixed point. Even if the formal elimination procedures
are very similar, the analysis is very different.
Handy references for the classical theory of Birkhoff normal forms
are \cite{SiegelM71, MeyerHO,Murdock,EFK13, EFK15}.
\subsection{Generic divergence both of the Birkhoff
Normal Form and the normalizing transformation.}
The BNF and the normalizing transformations are constructed as
formal power series.
The following natural questions are of great importance: the first one is whether the BNF
converges for Hamiltonians in a certain class.
The second---whether there is
a convergent normalizing transformation.
Concerning the first question, R. Perez-Marco \cite{PM} proved the following dichotomy: for any given nonresonant quadratic
part, either the BNF is generically divergent
or it always converges. The original proof was done in the setting of
Hamiltonian systems having
a non resonant elliptic fixed point. The extension of this result to the case of the torus, that
is not completely straightforward, has been worked out
by R. Krikorian, see Theorem 1.1 in \cite{Kri}.
Up to very recently it was unclear which of the possibilities is actually realized. A large progress has
been made by R. Krikorian \cite{Kri}, who proved that there exists a real analytic symplectic diffeomorphism $f$ of a two-dimensional annulus
such $f(\T \times \{0\})=(\T \times \{0\})$, $f(\theta,0)=(\theta+\om_0,0)$ with $\om_0$ Diophantine and having a
non-degenerate {\it divergent} Birkhoff normal form. Combined with the aforementioned result of Perez-Marco,
this implies that Birkhoff Normal Form of an analytic Hamiltonian is ``in general" divergent.
Concerning the normalizing
transformations, H. Poincar\'e proved that they are divergent for a
generic Hamiltonian.
C.~L.~Siegel proved the same statement in a neighborhood of an
elliptic fixed point (in fact, for a larger class of Hamiltonians than
just generic, \cite{Siegel54}).
This is implied by showing that the orbit structure of the map
in any neighborhood is
very different from that of the Birkhoff normal
form (which is integrable). Analogous results for
symplectic maps near an elliptic fixed point appear in \cite{Russmann59}.
Very different arguments showing divergence of normalizing transformations
for generic systems appear in \cite{Zehnder73} and for
some concrete polynomial mappings in \cite{Moser60}.
\subsection{Convergence of the transformations
under the Diophantine conditions for some particularly simple BNF}
There are classes of Hamiltonians for which we can guarantee the convergence of the normalizing transformation.
The following influential rigidity result was proved independently by A.~D.~Bruno \cite{Br71} and
H.~R\"ussmann \cite{Russmann67}. Note that the main assumption is
that the (in principle only formal) BNF is of a particular kind.
Consider an analytic Hamiltonian $H(I,\theta)$, whose
frequency $\lb_0$ satisfies a Diophantine condition \eqref{DC}.
Assume moreover that the Birkhoff normal form $N(I)$ of $H(I,\theta)$ is a {\it formal} function $B$ of one single variable
$\Lambda_0:=\< \lb , I\>$, i.e.,
$$
N(I)=B(\Lambda_0(I)).
$$
Then there exists an {\it analytic} normalizing
transformation, and the BNF is, in fact, analytic.
We remark that Bruno proves the above result under a weaker condition on $\lb_0$ than \eqref{DC}.
For analogous statements in the case of invariant tori see \cite{Br89}. Other modifications can be found in
\cite{Russmann02, Russmann04}.
This result has been recently generalised to a much more general
context by Eliasson, Fayad and
Krikorian \cite{EFK13, EFK15}.
We stress that in all these works mentioned above, $\lb_0$ is assumed to be non-zero and
the crucial assumption is that $\lb_0$ satisfies a Diophantine-type condition
and that the BNF is of a very simple form.
\subsection{``Sometimes'' convergence of the BNF implies
convergence of a normalizing transformation.}
Our main result is close in spirit to the above works,
but it {\it does not rely on a Diophantine condition}. In fact,
we consider a special class of diffeomorphisms
such that the frequency $\lb_0$ is zero. Thus,
the BNF is degenerate in the
previous sense. But within this class of Hamiltonians we
just use a standard non-degeneracy
assumption on the quadratic part. Namely, we prove the following.
\bigskip
\begin{theorem}\label{main}
Assume the following:
\begin{itemize}
\item[$(A_1)$] $H(I,\theta)$ has a formal Birkhoff normal
form $N(I)$ that starts with
quadratic terms in $I$, i.e
there exists a {\it formal} symplectic change of variables
$\Psi(I, \th)$, tangent to the identity, i.e.
$
\Psi(I,\theta)=(I+\cO^2(I),\phi+\cO(I))
$, such that
$$
H\circ \Psi (I,\th) = N(I)=N_0(I)+ \cO^3(I)
$$
in the sense of power series.
\item[$(A_2)$] $N_0(I)=I^{tr} \Omega I$ (for some symmetric $\Omega$) is
non-degenerate: $\det \Om \neq 0$.
\item[$(A_3)$] $N(I)=B(N_0(I))=N_0 + \sum_{j=2}^\infty b_j (N_0(I))^j$ where $B$ is an
analytic function.
\end{itemize}
Then there exists an invertible {\it analytic} symplectic transformation
$$
\Phi(I,\theta)=(I+\cO^2(I),\phi+\cO(I))
$$
such that
\begin{equation}\label{conjugacy}
H\circ \Phi (I,\th) = N(I).
\end{equation}
\end{theorem}
\bigskip
Note that we start from a resonant torus, so that the existence of a BNF of the form we assume, requires vanishing of (formal) obstructions.
Hence, our main result can be reformulated as saying that the formal assumptions imply convergence of the normalizing transformation.
Similar rigidity statements have appeared in other contexts. In
\cite[Ch. 5]{Poincare92}, H.Poincar\'e studied the
formal power series of canonical
transformations, which send a family of Hamiltonian
systems into a family of integrable systems (in the sense of
power series). In \cite{Poincare92}
it was shown that these formal
power series do not exist unless there are some conditions
(which
are not met in the three body problem for arbitrary masses).
The non-existence of
formal power series, a fortiori implies the non-existence of
analytic families of analytic transformations integrating the
three body problem.
The paper \cite{Ll} proved a converse to the result in
\cite{Poincare92}: if the system satisfies {a very specific
and generic} non-degeneracy condition,
then, existence of a formal power series that integrates the family
of transformations in the sense of power series implies existence of a convergent one.
Assumption $A_3$ is there for technical purposes, see Sec.~\ref{s_simplif}. Note that it is trivial for $d=1$. This assumption reminds of that
of R\"ussmann in \cite{Russmann67, Russmann02, Russmann04}.
The assumption that the Birkhoff normal
form is a function of $N_0$
has been discussed in \cite{Ga} under the name of
{\it relative integrability}. Two Hamiltonian dynamical
systems are
relatively integrable when one of them can be obtained
from the other by a symplectic change of coordinates and
a reparameterization of the time which only depends on
the total energy. That is, the orbit structures of the two systems in an
energy surface are equivalent up to a change of scale of
time. The paper \cite{Ga} includes several arguments for why
the notion of relative integrability is natural when discussing
formal equivalence.
In the present paper, however, the focus lies on the notion of
equivalence under a symplectic change of variables.
We show that, for a certain class of systems, equivalence in
the sense of formal power series implies equivalence
in the sense of analytic canonical changes of variables.
Hence, our main result can be understood as a rigidity result.
The class of systems for which this rigidity
result holds can be succinctly described as the set of systems that
relatively integrable with respect to the main term.
\medskip
In the context of formal equivalence implying analytically convergent
equivalence, it is natural to formulate:
\noindent{\bf Conjecture. }{\it Assume that an analytic Hamiltonian $H(I,\theta)$ as in \eqref{Ham1}
has a convergent BNF that satisfies the non-degeneracy assumption
that the frequency map is a local diffeomormphism.
Then there is
a convergent normalizing transformation. }
\medskip
Note that the problems studied in \cite{Russmann67} and \cite{Br71} do not satisfy
the hypothesis of the conjecture, even though they satisfy the conclusions.
In the other direction one can construct examples \cite{S} of
analytic maps near a hyperbolic
fixed point such that the Birkhoff normal form is quadratic (in the
above notations, $N=\Lambda_0$) with a
non-resonant set of eigenvalues, and any normalizing transformation to
the normal form diverges. In these examples,
the eigenvalues form carefully chosen Liouville vectors.
That is, the paper \cite{S} shows that, depending on the
Diophantine conditions, quadratic normal forms may be rigid or not.
The models in \cite{S} do not satisfy the hypothesis of
the conjecture above.
\subsection{Overview of the proof.}
The standard method of obtaining the Birkhoff Normal form is
an iterative procedure in which we construct the
transformations order by order: at the $n$-th step of the procedure
one computes the $n$-th order terms in the Taylor expansions,
assuming that all the terms of lower orders are computed.
It would appear natural to follow this scheme and
try to {\it estimate} the transformations at each step of the
recursive procedure. Unfortunately, this seems technically unfeasible.
One of the main complications in any possible proof of
convergence of the transformations is that even if
the BNF is unique, the formal transformations $\Phi_N$ are very far from
unique (Since the BNF depeds only on the actions, the $\Phi_N$
can be composed with any canonical
transformation which moves the angles but preserves the actions. So,
an essential ingredient of any proof of convergence should be
a especification of how to choose the normalizing transformations.
In this paper we use a
quadratically convergent method in which we double
the number of known coefficients at each step.
Roughly -- see more details in
the next paragraphs -- we will show that if the formal obstructions
vanish we can choose a sequence of canonical transformations
that proceed to converge quadratically: doubling the order of
the BNF at every step of the construction. More importantly,
there is a specific choice of the transformation that satisfies
very explicit bounds. The bounds on the new transformation in terms
of the remainder turn out to involve a loss of derivatives. Therefore
we need to implement a Nash-Moser scheme to estimating the important objects in a sequence of
domains which decrease slowly.
Here is a short overview of the proof; all the necessary notations are introduced in the next
section. At the $n$-th step of the iterative procedure we will start with a
Hamiltonian of the form
$$
H_n(I,\th)=N_n(I)+ \widetilde{R_n}(I,\th),
$$
where $N_n(I)$ is a polynomial in $I$ of degree $m_n=2^n+1$, and the remainder term $\widetilde{R_n}$ is small in the following sense:
for a certain domain-dependent norm, introduced in Sec. \ref{sec_norms}, for a certain small $\dt_n$ (we assume $\dt_n \to 0$
with $n\to \infty$) and $\kp>0$
the remainder term satisfies
$|\widetilde{R_n}|_{\rho_n,\rho_n}\leq \delta_{n}^\kp $.
At this step we construct a symplectic change of coordinates $\Phi_n$,
such that
$$
H_n\circ\Phi_n (I,\th)=N_{n+1}(I)+ \widetilde{R_{n+1}}(I,\th),
$$
where $N_{n+1}$ has degree $m_{n+1}=2m_n-1$, and
$|\widetilde{R_{n+1}}|_{\rho_{n+1},\rho_{n+1}}\leq \delta_{n+1}^\kp
=2^{-\kp}\delta_{n}^\kp$.
We construct $\Phi_n$ as a time one map of a the flow of
a Hamiltonian vector field $F_n$.
The main ingredient consists in constructing and estimating the norm
of $F_n$ (and thus $\Phi_n$), which is found as a solution of a
certain homological equation (see \eqref{eq_homol} and in a
simplified form \eqref{homol-eq}). In general,
this equation may not have even a formal solution
unless some constraints are met. However, the assumption of Theorem
\ref{main} implies that this equation does have a formal solution.
The key observation in this paper is the following:
if this homological equation has a {\it formal solution}, then it also
has an {\it analytic solution with tame
estimates} for it
(in the sense of Nash-Moser theory). This statement is the contents of
Lemma \ref{lemma-homol-eq0}.
We note that the tame estimates use an argument different from the
matching of powers.
The procedure can be repeated, because the main assumption used
to show the existence of solutions of the Newton equation is
that there is a formal solution to all orders. This assumption
is clearly preserved if we make any analytic change of variables.
Once we know that the Newton procedure can be repeated infinitely
often, the convergence is more or less standard.
\section{Notations and a step of induction. }
\subsection{ Notations.}
\subsubsection{Norms and majorants.} \label{sec_norms}
Let $\T^d=\R^d /\Z^d$ be a $d$-dimensional torus, and for $\si>0$ consider its complex extension
$\T^d_\si =\left(\R^d+(-\si,\si)\sqrt{-1} \right)/ \Z^d$. Let $\D^d_{\rho }=\{I\in
\C^{d}: |I|< \rho \}$ be a complex disc, and define the "d-dimensional annulus"
$$
\A_{\rho,\si }:=\D_{\rho}^d\times \T^d_\si .
$$
Let $\cO(\A_{\rho,\si })$ be the set of functions holomorphic in $\A_{\rho,\si }$ that are real symmetric, i.e., such that
$\overline{f(\bar I,\bar \th)}=f(I, \th)$ (where the bar stands for the complex conjugate).
We use supremum norms over $\A_{\rho,\si }$, denoted by $\|f\|_{\rho,\si
}$. In the same way we define the set $\cO(\D_{\rho })$
with the corresponding norm
$\|f\|_{\rho }$ being the sup-norms over the disc
$\D^d_{\rho }$.
For a function $f\in \cO(\A_{\rho,\si })$ consider its Taylor-Fourier representation in the powers of $I$:
$
f(I,\th)=\sum_{j\in \N^d} \sum_{k\in \Z^d} f_{j,k}e^{2\pi i
\<k,\theta \>} I^j
$.
Consider a majorant for $f$ of the form
$$
\widehat f(I) = \sum_{j\in \N^d} \sum_{k\in \Z^d} |f_{j,k}| I^j e^{2\pi|k|\si}
$$
We denote
by $|f|_{\rho,\si }$ the norm of the corresponding majorant $\widehat f(I)$:
$$
|f|_{\rho,\si }=\|\widehat f\|_{\rho,\si }.
$$
Clearly, $\|f\|_{\rho,\si } \leq |f|_{\rho,\si }$. Analogous notation
$|f|_{\rho}$ corresponds to the norm $\|f\|_{\rho}$ above.
In what follows we will mostly have $\si =\rho$.
\subsubsection{Important constants for the iterative procedure.}\label{not_const}
\begin{itemize}
\item Let $\rho_0=\min \{1, \rho\}$,
\item The order of polynomials involved in the $n$-th step of the iterative procedure is
$$
m_{n} =2^n +1.
$$
\item The norm of the rest term $\widetilde{R_n}$ at the $n$-th step will be
estimated as $|\widetilde{R_n}|_{\rho_n}\leq \delta_{n}^\kp $.
Let
$$
\begin{aligned}
&\kappa = d + 6, \\
&b = 2^{-(\kappa + 3)} \\
& \delta_0= \rho_0 b 2^{-3} = \rho_0 2^{-(\kappa + 6)}\\
& \delta_{n+1}= 2^{-1} \delta_{n}.
\end{aligned}
$$
\item Finally, let
$$
q_n=(2b)^{2^{-(n+1)}},
$$
and
$$
\rho_{n+1}=(\rho_n-3\dt_n)q_n.
$$
\end{itemize}
\subsubsection{Polynomials.}\label{not_poly}
In the iterative procedure we will work with polynomials in
$I$ whose coefficients depend on $\th$.
\begin{itemize}
\item Let
\begin{equation}\label{def_N0}
N_0(I)=I^{tr} \Omega I
\end{equation}
where $\Omega$ is
a symmetric non-degenerate matrix: $\det \Om \neq 0$.
\item An expression $M=f(\th) I^k $ (where $k$ is a multi-index) is called a monomial.
\item We will say that a monomial $M_{k,l}=I^ke^{2\pi i \< l,\th\>}$ is {\it resonant} if it satisfies $\{N_0, M\}=0$.
\item $R^{[j]} (I,\th)$ stands for a homogeneous polynomial in $I$ of
degree $j$ with coefficients depending on $\th$:
$$
R^{[j]}(I,\th)=\sum_{|k |=j} r_{k }(\th ) I^{k }.
$$
\item We also use notation $R^{[m,n]}$ to denote the range of degrees in $I$:
$$
R^{[m,n]} (I,\th)=\sum_{j=m}^n R^{[j]} (I,\th), \quad R^{[\geq m]} (I,\th)=\sum_{j=m}^\infty R^{[j]} (I,\th).
$$
\end{itemize}
Let $m_n$ be as above. The following functions will be of special importance.
\begin{itemize}
\item The normal form $N(I)$ is assumed to have the form
\begin{equation}\label{N=B}
N(I) = B(N_0(I))= N_0(I) + \sum_{j=2}^\infty b_j (N_0(I))^j.
\end{equation}
Denote
\beq\label{def_N}
N_n=N^{[2,m_n]}= \left( B(N_0)\right)^{[2,m_n]};
\eneq
in particular, since $m_0=2$, $N_0=N_0^{[2,m_0]}=N_0^{[2]}$ is quadratic.
\item The rest term at the $n$-th inductive step is $\widetilde{R_n}(I, \theta)$:
\beq\label{def_tR}
\widetilde{R_n}= \widetilde{R_n}^{ [>m_{n} ] }.
\eneq
\item We will also need polynomials in $I$ with $\theta$-dependent coefficients:
$R_n(I, \theta)$ and $F_n(I, \theta)$ of the following degrees:
\beq
R_n= R_n^{ [m_n+1,m_{n+1} ] }, \quad F_n=F_n^{[m_n, m_{n+1}-1]}.
\eneq
\end{itemize}
\subsection{Base of induction: an equivalent problem.}
\begin{Lem}\label{lemma_base}
Suppose that
$$
H(I,\th)=N_0(I)+ \widetilde {R_0} (I,\theta) \in
\cO(\A_{\rho,\si }),
$$
where $|\widetilde{R_0}|_{\rho,\si }\leq \delta$, and there exists a
formal (resp., analytic) symplectic transformation
$$
\Psi(I,\theta)=\left(\phi(I,\theta),\,
\psi(I,\theta) \right)=(I+\cO^2(I),\theta + \cO(I))
$$
such that
$$
H\circ\Psi(I,\th)= N(I)=N_0(I) + \sum_{j=2}^\infty b_j (N_0(I))^j .
$$
Then for any $a > 0$ there exists a Hamiltonian $\widehat
H(I,\th)$ and a formal (resp., analytic) symplectic transformation
$
\widehat \Psi(I,\theta)=(I+\cO^2(I),\theta + \cO(I))
$ such that
$$
\widehat H\circ \widehat \Psi(I,\th) =N_0(I)+ \widehat {R_0} (I,\theta) \in
\cO(\A_{\frac1{a} \rho,\si }),
$$
where $| \widehat {R_0}|_{\frac1{a} \rho,\si }\leq a \delta$, and
$$
N(I)=N_0(I) + \sum_{j=2}^\infty b_j a^{2(j-1)}(N_0(I))^j .
$$
\end{Lem}
\bg \noindent {\it Proof. \ }\bk Define $\widehat
H(I,\th)=\frac1{a^2}H(aI,\th)$, and
$
\widehat \Psi(I,\theta)= \left( \frac{1}{a}\phi(aI,\theta),\,
\psi(aI,\theta) \right)
$.
It can be verified directly that $\widehat \Psi$
is symplectic and tangent to the identity. Moreover,
$$
\widehat H \circ \widehat \Psi(I,\theta)= \frac{1}{a^2}
H(\phi(aI,\theta),
\psi(aI,\theta)) =
N_0(I) + \sum_{j=2}^\infty b_j a^{2(j-1)}(N_0(I))^j.
$$
\hfill {\bg $\Box$}
\bigskip
\subsection{Induction step.}
While the base of induction is given by formula \eqref{est_R0},
the step of the iterative procedure is provided by the following
proposition.
\begin{proposition}\label{lemma1}
For a fixed $n > 0$, let $m_n$, $\rho_n$ and $\dt_n$ be as in Sec. \ref{not_const} above.
Suppose that $H_n(I, \theta)$ is formally conjugated to the BNF of the
form \eqref{N=B}:
$$
N(I)=N_0(I) + \sum_{j=2}^{\infty} b_j (N_0(I))^j,
$$
and the normal form satisfies:
\begin{equation}\label{est_N}
|N^{[ m_n+j ]}|_{\rho_n}<\dt_n^{\kappa+1}, \quad j=0, \dots m_n,
\end{equation}
and denoting
$
g_{2j}(I) = jb_j (N_0(I))^{j-1}
$,
we assume
\begin{equation}\label{est_g}
|g_j|_{\rho_n} \leq \frac1{4^j}, \quad j=1,\dots ,m_n ;
\end{equation}
Suppose that
$$
H_n(I, \theta) = N_n(I) + \widetilde{R_n} (I, \theta),
$$
where $N_n(I) = \left( B(N_0(I))\right)^{[2,m_n]} $
and $\widetilde{R_n}= \widetilde{R_n}^{ [>m_{n} ] } $ satisfies
$$
|\widetilde {R_n} |_{\rho_n,\rho_n} \leq \delta_n^{\kappa}.
$$
Then there exists a symplectic change of coordinates $\Phi_n:(I',
\theta')\mapsto (I,\theta)$,
$$
\Phi_n(I', \theta')=(U^{(n)} (I', \theta'), V^{(n)}(I', \theta')),
$$
given by a Hamiltonian $F_n=F_n^{[m_{n},m_{n+1}-1]}$ such that
\beq\label{eqHn}
H_{n+1}(I', \theta'):=H_n \circ \Phi_n(I', \theta') = N_{n+1} (I') +\widetilde { R_{n+1} }(I', \theta'),
\eneq
where $N_{n+1}(I')=N^{[2,m_{n+1}]}(I')$, $\widetilde { R_{n+1}}(I',\theta')=\widetilde {R_{n+1}}^{[>m_{n+1}]}(I',\theta')$, and
\beq\label{estRn}
|\widetilde { R_{n+1}} |_{\rho_{n+1},\rho_{n+1}} \leq \dt_{n+1}^{\kappa} .
\eneq
Moreover, $\Phi_n(I', \theta')=(U^{(n)}(I', \theta'), V^{(n)}(I',
\th'))$ satisfies
\beq\label{estPhi}
\sum_{j=1}^d \|U_j^{(n)}(I', \theta')-I'_j \|_{\rho_n-3\dt_n,\rho_n-3\dt_n} +
\|V_j^{(n)}(I', \theta')-\theta'_j \|_{\rho_n-3\dt_n,\rho_n-3\dt_n} < \dt_n,
\eneq
and the inverse map,
$\Phi_n^{-1}(I, \theta):=(U^{(-n)}(I, \theta),V^{(-n)}(I, \theta))$, satisfies
\beq\label{estPhi-1}
\sum_{j=1}^d \| U_j^{(-n)}(I, \theta)-I_j \|_{\rho_n-3\dt_n,\rho_n-3\dt_n} +
\| V_j^{(-n)}(I, \theta)-\theta_j \|_{\rho_n-3\dt_n,\rho_n-3\dt_n} < \dt_n.
\eneq
\end{proposition}
The proof of this proposition constitutes the
main technical tool of this paper. It
implies Theorem 1 in a standard way. See, e.g., \cite{Russmann67},
pp 61-63). For convenience, we give a proof below.
\subsection{Proof of Theorem 1.}
Lemma \ref{lemma_base} permits us to assume without loss of
generality that for the given Hamiltonian
$
H_0(I,\th):=H(I,\th) = N_0(I)+\widetilde{R_0}(I,\th),
$
\begin{equation}\label{est_R0}
|\widetilde{R_0}|_{\rho_0,\rho_0}\leq \delta_{0}^\kp.
\end{equation}
Since the function $B$ is analytic, the same lemma permits us to assume that \eqref{est_N} and \eqref{est_g} hold for each $n$.
The step of induction is provided by Proposition \ref{lemma1}.
Since $H_{n}$ is formally reducible to the normal form $N$, the same can be
said about $H_{n+1}$.
Repetition of this process leads to a sequence of transformations
$$
T_n=\Phi_0\circ\Phi_1\circ\dots\circ\Phi_{n-1}.
$$
Let us show that $T_n$ converges to
the desired coordinate change $\Phi=T_\infty$, analytic in the polydisc
$\A_{\rho_\infty,\rho_\infty}$, where $\rho_0 b < \rho_\infty < \rho_0$. Indeed, with the notations of Sec. \ref{not_const},
$$
3 \sum_{k=0}^\infty \dt_k \leq 3\cdot 2\dt_0 < 3\cdot 2 \rho_0 b
2^{-3} <\rho_0 b .
$$
Then
for any $n$ we have
$$
\rho_{n+1}=q_n (\rho_{n} - 3\dt) \geq \rho_{0} \prod_{j=0}^n q_j - 3\sum_{j=0}^n \dt_n
\geq \rho_{0} \prod_{j=0}^\infty q_j - 3\sum_{j=0}^\infty \dt_n\geq \rho_{0} 2b - 3\cdot 2\dt_0 >b\rho_{0}
$$
It is left to prove that $T_n$ converges of to an analytic function
$T_\infty$, satisfying \eqref{conjugacy}.
Denote the variables, involved in the $n$-th step of the induction by
$w_{n-1}=(I,\theta)$ and $w_{n}=(I',\theta')$, where
$$
w_{n}=\Phi_{n-1}^{-1} w_{n-1}.
$$
In these notations,
$$
w_{0}=\Phi_{0} \circ \Phi_1 \circ \dots \circ\Phi_{n-1}w_n = T_n w_n.
$$
Now, for $w_{n}=(I',\theta')$ we have
$$
H\circ T_n(I',\theta')= N_n(I')+\widetilde{R_n}(I',\theta').
$$
Since $(\Phi_{n}(I',\theta')-(I',\theta'))$
starts with the terms of degree $2^n$ in $I'$, for each $j$ the expansion of
$(T_n(I',\theta')-T_{n+j}(I',\theta'))$ starts with the terms of
degree $2^n$ in $I'$.
This implies that
the sequence of maps $T_n$ formally converges, when $n\to \infty$, to
a formal map $T_\infty$ such that \eqref{conjugacy} holds:
$$
H\circ T_\infty (I',\theta')= N(I').
$$
We still need to show that $T_\infty$ is analytic. It is more convenient to prove that the maps
$$
T_n^{-1}:=\Phi_{n-1}^{-1}\circ\dots\circ\Phi_1^{-1}\circ\Phi_{0}^{-1}
$$
converge to an analytic map $T_\infty^{-1}$.
By Proposition \ref{lemma1}, the map
$$
w_{n+1}=\Phi_n^{-1} w_n
$$
is analytic in $\A_{\rho_0 b/2,\rho_0 b/2}$, and for all $n$ we have:
$$
|\Phi_n^{-1} w_n- w_n|_{\rho_0 b/2,\rho_0 b/2}\leq \dt_n,
$$
since $\rho_n-3\dt\geq \rho_{n+1}> \rho_0b$ for all $n$.
Therefore, the map $T_n^{-1}$ such that
$$
w_{n}=T_n^{-1} w_0
$$
is analytic in $\A_{\rho_0 b/4,\rho_0 b/4}$, and for such $w_0$ we have
$$
|T_n^{-1} w_0|\leq \sum_{j=0}^{n-1} |T_{j}^{-1}(w_j)-w_j| +|w_0|\leq
\sum_{j=0}^{\infty}\dt_j +\rho_0b/4 \leq \rho_0b/2.
$$
Estimate
$$
|T_{n+m}^{-1}(w_0)-T_n^{-1}(w_0)|_{\rho_0 b/4,\rho_0b/4}\leq
\sum_{j=n}^{n+m-1} |T_{j}^{-1}(w_j)- w_j)|_{\rho_0 b/4,\rho_0b/4}\leq
\sum_{j=n}^{\infty}\dt_j =2^{1-n}\dt_0
$$
implies the convergence of the sequence of maps $T_n^{-1}$ to an
analytic map $T_\infty^{-1}$ in $\A_{\rho_0 b/4,\rho_0 b/4}$.
Since the formal inverse of $T_\infty^{-1}$ is the series $T_{\infty}$, the latter also defines an analytic function,
providing the desired coordinate change. We set $\Phi=T_\infty$ in the notations of Theorem \ref{main}.
\hfill {\bg $\Box$}
\section{Formal analysis.}
Here we start the proof of Proposition \ref{lemma1} by the formal analysis of the
iterative procedure.
\subsection{Iterative Procedure.}
Given $H_n$ as in Proposition \ref{lemma1},
we will construct $\Phi_n$ as the time one map of the flow of a
Hamiltonian $F_n$, i.e., $\Phi_n = X_{F_n}^1$ where $X_{F_n}^t$ is
the flow
defined by
$$
\dot I =F_\th(I,\th),\quad \dot \th =-F_I (I,\th).
$$
In this case, $\Phi_n$ is automatically symplectic.
Notice that the normalising transformation $\Phi_n$, as well as the
corresponding generating function $F_n$, is not unique (one can
compose with rotations in the angles which preserve the actions, for
example). Clearly, the transformation that converges has to be
very carefully chosen.
In the following Lemma~\ref{lem_help} we show
that if a (formal) normalizing transformation exists, then there
exists (another) normalizing transformation of a special kind. Namely, such that the corresponding generating function
is a polynomial (in the sense of section \ref{not_poly}),
$F_n=F_n^{[m_n, m_{n+1}-1]}$ and free from resonant monomials (see notations in Sec. \ref{not_poly}).
The idea of the proof is that we can always move the formal normalizing
transformation by composing with some transformations that
do not change the normal form. Therefore, we can ensure that the
normalizing transformations belong to a space which is transversal
to the space spanned by resonant monomials.
Note that in the proof of Lemma~\ref{lem_help}
we use crucially the fact that the normal form is
a function of $N_0$ so that the resonant terms are the same at all orders.
There are some analogies between Lemma~\ref{lem_help} and
Proposition 2.6 in \cite{Ll}, but that result is significantly less
delicate since there is an extra parameter that controls the smallness.
In our case, the variable $I$ controls both the smallness and the distance
to the origin at the same time.
Let $\{\cdot, \cdot\}$ denote the standard
Poisson bracket. Recall that
for a differentiable function $G$ it holds:
$$
\frac{d}{dt}G\circ X_F^t =\{ G,F\}\circ X_F^t.
$$
\bigskip
\begin{Lem}\label{lem_help}
Suppose that for $H(I,\th)$
there exist $N_{2m}(I)=N_0 + B(N_0)$ with $B(X)=\sum_{j=2}^{m} b_j X^{j}$, $R(I,\th)=R^{[> 2m]}(I,\th)$ and $G(I,\th)=\mathcal O^2(I) $ such that
$\Psi:=X_G^1$ satisfies
\[
H\circ \Psi (I,\th)=N_{2m}(I)+R(I,\th).
\]
\def\tR{{\tilde R}}
\newcommand{\dbtilde}[1]{\accentset{\approx}{#1}}
\def\ttR{{\dbtilde {R}}}
\def\tPsi{{\tilde \Psi}}
\def\tG{{\tilde G}}
\def\cL{{\mathcal L}}
\begin{enumerate}
\item Then there exists $\tG(I,\th)$, that is free from resonant monomials of order $< 2m$,
such that $\tilde \Psi :=X_{\tilde G}^1$ normalises $H$ to the same normal form, i.e.,
for some
$\tR(I,\th)=(\tilde R)^{[> 2m]}(I,\th)$ we have:
$$
H\circ \tPsi(I,\th)=N_{2m}(I)+\tR (I,\th).
$$
\item If, an addition to the previous assumption, we have that the original $H(I,\th)$ has the form
$$
H(I,\th)= N_{m}(I)+ R^{[>m]}(I,\th),
$$
where $N_{m}=N_{m}^{[2,\dots , m]}$, then there exists a polynomial $F=F^{[m, 2m-2]}$, that is free from resonant monomials,
such that $\Phi :=X_{F}^1$ normalises $H$ to the same normal form, i.e., for some
$\ttR(I,\th)=\ttR^{[> 2m]}(I,\th)$ we have:
$$
H\circ \Phi (I,\th)=N_{2m}(I)+\ttR(I,\th).
$$
\end{enumerate}
\end{Lem}
\bg \noindent{\it Proof: (1).} \bk
All the calculations below are made in the sense of formal Taylor-Fourier expressions.
Suppose that $K(I,\th)$ is such that $\{N_0, K\}=0$. Notice that in this case
$\{N_{2m}, K\}= B' (N_0)\{N_0, K\} =0$.
Use $K(I,\th)$ as a Hamiltonian to define $k(I,\th):=X_{K}^1$.
Then by Taylor formula we have:
$$
\begin{aligned}
H\circ \Psi \circ k & = (N_{2m}+R )\circ k= (N_{2m} + R )\circ
X_{K}^t \Big|_{t=1} = N_{2m} + R + \{(N_{2m}+R), K\} \\
&+ \frac12\{\{(N_{2m}+R), K \} ,K\}+ \dots =
N_{2m}+R_1,
\end{aligned}
$$
where $R_1(I,\th)=R_1^{[> 2m]}(I,\th)$.
It is a classical fact that the composition $\Psi \circ k$
in the sense of formal power series is the time-one
map of another Hamiltonian given by the Cambell-Baker-Dynkin
formula \cite[Appendix C]{Dragt}, \cite[Appendix]{LlMM}; here we denote it by CBD formula.
Note that in these references the usual notation for the Hamiltonian vector field defined by $G$ is $\cL_G$, and $\exp(\cL_G)$ stands for its time one map.
In the present paper the same map is denoted by $X_G^1$. Now, suppose that
$\Psi = X_G^1$ and $ k = X_K^1$.
CBD formula implies that the composition of these maps satisfies:
\[
\begin{split}
& \tilde \Psi :=\Psi \circ k = X_{\tilde G}^1 , \quad \text{where}\\
& {\tilde G} = G + K + \frac{1}{2}\{G, K\}
+ \frac{1}{12}\{G, \{G, K\}\}
- \frac{1}{12}\{K, \{K, G\}\} + \cdots
\end{split}
\]
The last sum is to be understood in the sense of formal power series in $I$.
To prove Lemma \ref{lem_help}, we use CBD formula, and choose $K$ recursively (order by order in $I$) so that ${\tilde G} $
has no resonant terms up to order $2m$. At each step of the recursion we choose $(-K(I,\theta))$
to be equal to the lowest order resonant term of $G$, and set ${\tilde G} $ to be the new $G$. As we saw above, the map $\tilde \Psi =\Psi \circ K$, used as a normalization map, brings $H$ to the same normal form as $\Psi$ did. But its generating Hamiltonian ${\tilde G} $ has no lower order resonant monomials. Iterating this procedure, we get a normalization with the desired property.
\medskip
\bg \noindent{\it (2).} \bk
Since we can normalise $H=N_{m}+R^{[>m]} $ to $N_{2m}$ with the help of the generating function
$G=\mathcal O^2(I)$, then, by {\it (1)}, we can also achieve the
normalization using the transformation $\tilde \Psi$ generated by a resonance-free Hamiltonian
$\tilde G $.
Note that $\tilde G =\mathcal O^2(I)$.
By the Taylor formula for power series, we have:
\[
\begin{aligned}
H\circ \tilde \Psi & = (N_{m}+R^{[>m]} ) \circ {\tilde \Psi} = (N_{m} + R^{[>m]} )\circ
X_{\tilde G}^t \Big|_{t=1} = N_{m} + R^{[>m]} \\
&+ \{(N_{m}+R^{[>m]}), \tilde G\}
+ \frac12\{ \{(N_{m}+R^{[>m]}), \tilde G \}, \tilde G \}+ \dots
=
N_{2m}+R_1.
\end{aligned}
\]
Since $\tilde G $ is resonance-free, any monomial $P$ in $\tilde G $ gives a non-zero impact $\{N_0, P\} $ to the sum above, whose order in $I$ is strictly larger than the order of $P$.
By comparing the orders of the coefficients in $I$ we see that the lowest possible order of a monomial in
$\{N_{0}, \tilde G\} $ is the same as that in $R^{[>m]}$, and hence $\tilde G=\tilde G^{[\geq m]}$.
Finally notice that the reduced
generating function $F:=\tilde G^{[m, 2m-2]}$ produces the same normal form.
\hfill {\bg $\Box$}
\bigskip
The following lemma introduces the notations used in the proof of the Main Theorem. Here we use the results of Lemma \ref{lem_help} to relate the conjugating function to the
solutions of the homological
equation \eqref{eq_homol} below.
\medskip
\begin{Lem}\label{lem_homol_eq}
Adopt the notations for the degrees of polynomials from
Sec. \ref{not_poly} (in particular, $N_n=N^{[2,m_n]}$ as in \ref{def_N}, and $R_n=R_n^{ [m_n+1,m_{n+1} ] }$).
Let $B(X)=\sum_{j=1}^{\infty} b_j X^{j}$. Suppose that $H_n$ has the form
$$
H_n= N_n +\widetilde{ R_n} = N_n + R_n +\widetilde{ R_n}^{[>m_{n+1}]}.
$$
where $N_{n}(I)=N_0 + B(N_0)^{[4,m_n]}$.
Suppose that there exists $G(I,\th)=\mathcal O^2(I) $ such that
$\Psi:=X_G^1$ satisfies
\[
H\circ \Psi (I,\th)=N_{m+1}(I)+R(I,\th).
\]
Then there exists a polynomial (in $I$)
$F_n=F_n^{ [m_n,m_{n+1}-1] }$ with the following properties:
the time one map $\Phi_n:= X_{F_n}^1$ satisfies
$$
H_{n+1}:=H_n \circ \Phi_n = N_{n+1} + \widetilde{ R_{n+1}},
$$
and $F_n$ satisfies
\begin{equation}\label{eq_homol}
\{N_n, F_n\}^{ [m_n+1,m_{n+1}] }+ R_n + N_n - N_{n+1} =0 ,
\end{equation}
and
$$
\widetilde{ R_{n+1}}:= A_n+B_n +C_n,
$$
where
\begin{equation}\label{not_AnBn}
A_n:= \widetilde{R_n}^{[>m_{n+1}]}\circ \Phi_n , \quad
B_n:=\int_0^1\{ (1-t) \{N_n, F_n \}+R_n,F_n\}\circ X_{F_n}^t dt,
\end{equation}
\begin{equation}\label{not_Cn}
C_{n} = (\{N_n, F_n \} )^{[>m_{n+1}]} .
\end{equation}
\end{Lem}
Notice that the expressions for $A_n$, $B_n$, $C_n$
start with terms of order $m_{n+1}+1$, and hence, $\widetilde{ R_{n+1}}=\widetilde{ R_{n+1}}^{[>m_{n+1}]}$, as needed.
\medskip
\bg \noindent{\it Proof.} \bk
Let $m=m_n=2^n+1$. Then $m_{n+1}= 2m-1$. With the notations for the degrees of polynomials from
Sec. \ref{not_poly},
Lemma \ref{lem_help} implies that there exists a polynomial $F_n=F_n^{ [m_n,m_{n+1}-1] }$
such that $\Phi_n:= X_{F_n}^1$ satisfies $H_n \circ \Phi_n = N_{n+1} +\widetilde{R_{n+1}}$.
By the Taylor formula we have:
\beq\label{eq1}
\begin{aligned}
H_n \circ \Phi_n = &(N_n + R_n +\widetilde{ R_n}^{[>m_{n+1}]})\circ
X_{F_n}^t \Big|_{t=1} = N_n + \{N_n, F_n\} + R_n +\\
&\int_0^1\{ (1-t)
\{N_n, F_n \}+R_n,F_n\}\circ X_{F_n}^t \, dt + \widetilde{R_n}^{[>m_{n+1}]}\circ \Phi_n \\
= &N_{n+1} +\widetilde{R_{n+1}}.
\end{aligned}
\eneq
Notice that by extracting all the terms of orders $m_n+1,\dots ,m_{n+1} $ from the equation above, one gets the cohomological equation \eqref{eq_homol}.
\hfill {\bg $\Box$}
\bigskip
\subsection{Homological equation order by
order.}\label{s_order_by_order}
Here we rewrite equation \eqref{eq_homol} as a (finite) set of
equations for each degree of $I$. Equations corresponding to degrees
$m_n+1,\dots , m_{n+1}$ will formally determine $F_n$ (they are written out explicitly
in (\ref{homol-eq-by-order})). The rest of equations define $C_n$
(which is a part of the new remainder term).
Equating coefficients with the same homogeneous degree in $I$
in both sides of \eqref{eq1} we obtain for the degrees from $m_n+1$ to
$m_{n+1}$
the following recursive formula (we write $m$ instead of $m_n$ for
typographic reasons):
\beq\label{homol-eq-by-order}
\beal
&\{N_0, F^{[m]} \}+R^{[m+1]} =N^{ [m+1] }, \\
&\{N_0, F^{[m+1]} \} + \{N^{[3]}, F^{[m]} \}+ R^{[m+2]}= N^{[m+2]}, \\
&\{N_0, F^{[m+2]} \} + \{N^{[4]}, F^{[m]} \}+ \{N^{[3]}, F^{[m+1]} \}+ R^{[m+3]}= N^{[m+3]}, \\
&\dots \\
&\{N_0, F^{[2m-2]} \}+ \sum_{j=0}^{m-3}\{N^{[m-j]}, F^{[m+j]} \} + R^{[2m-1]}=N^{[2m-1]}.
\enal
\eneq
Recall that $2m_n-1=m_{n+1}$, see Sec. \ref{not_const}. From the formal
solvability
we know that each of these equations has
a formal solution $F_n^{[m+j]}$. Of course, such a solution is not
unique. We will make the solution unique by prescribing the condition
$$
\int_{\T^d} F_n^{[m+j]}(I,\th) =0.
$$
As we will see, this normalization will allow us to get the estimates needed for the proof of the convergence.
The sum of the terms of orders $m_{n+1}+1, \dots , m_{n+1}+m_n-2$ (i.e., $2m_{n}, \dots , 3m_n-3$) that appear
in equation (\ref{homol-eq}) is denoted by $C_n$. In the notation $m=m_n$, we have: $C_n=C_n^{[2m, 3m-3]}$.
The terms of the uniform degree satisfy
\beq\label{C-by-order}
\beal
&C_n^{[2m]}=\{N_{[3]}, F^{[2m-2]} \}+ \{N_{[4]}, F^{[2m-3]} \} + \dots + \{N_{[m]}, F^{[m+1]} \}, \\
&C_n^{[2m+1]}=\{N_{[4]}, F^{[2m-2]} \}+ \{N_{[5]}, F^{[2m-3]} \} + \dots + \{N_{[m]}, F^{[m+2]} \} \\
&\dots \\
&C_n^{[3m-3]}=\{N_{[m]}, F^{[2m-2]} \}.
\enal
\eneq
This can be written more compactly as
\beq\label{def_Cn}
C_n= \sum_{k=1}^{m-2} \{F^{[2m-1-k]}, \sum_{j=k+2}^{m}N^{[k+j]} \} .
\eneq
This should be viewed as a definition of the remainder term
$C_n$.
\subsection{An important simplification.}\label{s_simplif}
In the case when the {\it normal form is an analytic function of
$N_0(I)$} as in \eqref{N=B},
we have an important simplification.
Denote
\beq\label{notation_gj}
g_{2j}(I):= j b_{j} (N_0(I))^{j-1} \text{ and }
g_{2j+1}(I)\equiv 0.
\eneq
Then for $j\in \N$ we have:
\beq\label{def_g_j}
\begin{aligned}
&\{N^{[2j]}, F\} = \{b_{j} (N_0)^{j}, F\}= j b_{j} (N_0)^{j-1} \{ N_0, F
\}=
g_{2j}(I) \{ N_0, F \} \\
&\{N^{[2j+1]}, F\}= g_{2j+1}(I) \{ N_0, F \}\equiv 0.
\end{aligned}
\eneq
We formulate this as a lemma:
\blm\label{l_simplification}
If the normal form is an analytic function of
$N_0(I)$ as in \eqref{N=B},
then equation \eqref{homol-eq-by-order} is equivalent to
\beq\label{homol-eq-by-order_s}
\beal
&\{N_0, F^{[m]} \}+R^{[m+1]} =N^{ [m+1] }, \\
&\{N_0, F^{[m+1]} \} + g_{3}(I)\{N_0, F^{[m]} \}+ R^{[m+2]}= N^{[m+2]}, \\
&\{N_0, F^{[m+2]} \} + g_{4}(I) \{N_0, F^{[m]} \}+ g_{3}(I)\{N_0, F^{[m+1]} \}+ R^{[m+3]}= N^{[m+3]}, \\
&\dots \\
&\{N_0, F^{[2m-2]} \}+ \sum_{j=0}^{m-3}g_{m-j}(I)\{N_0, F^{[m+j]} \} + R^{[2m-1]}=N^{[2m-1]}.
\enal
\eneq
\elm
and
\beq\label{def_Cn_G}
C_n= \sum_{k=1}^{m-2} \left( \{F^{[2m-1-k]}, \, N_0 \} \cdot \sum_{j=k+2}^{m} g_j \right).
\eneq
\subsection{Homological equations in majorants.} \label{s_majorants}
Here we study a simple recursive formula and estimate its terms.
Later it will provide an important estimate of $|\{N_0, F^{j}
\}|_{\rho_n,\rho_n}$. Here is the idea: suppose that in the Lemma
above for some
$\eps>0$, for all $j=0,\dots ,m$ we
have:
$$P_j:=|R^{[m+j]}|_{\rho_n,\rho_n} +
|N^{ [m+j] }|_{\rho_n,\rho_n}\leq \eps, \quad |g_{j}|_{\rho_n}\leq 1/4^j.$$
Define $S_j$ by the relations \eqref{homol-eq-by-order1} below.
Then, by Lemma \ref{l_simplification}, for all $j=0,\dots ,m$ we have
$$
|\{N_0, F^{j} \}|_{\rho_n,\rho_n}\leq S_j.
$$
\blm\label{l_est_NF}
Given $ \eps >0$,
suppose that for all $j=1, \dots ,m-1$ the numbers
$P_j$ satisfy
$$
0< P_j \leq \eps .
$$
Let $S_j$ be defined recursively by equations
\begin{equation}\label{homol-eq-by-order1}
\begin{aligned}
&S_{1} = P_{1}, \\
&S_{2} = P_{2} + \frac14 S_{1} , \\
&S_{3} = P_{3} + \frac14 S_{2}+ \frac1{4^2} S_{1} \\
&S_{4} = P_{4} + \frac14 S_{3}+ \frac1{4^2} S_{2}+ \frac1{4^3} S_{1} \\
&\dots \\
&S_{m-1} = P_{m-1} + \sum_{j=1}^{m-1} \frac1{4^j} S_{m-1-j} .\\
\end{aligned}
\end{equation}
Then for each $j$ we have
$$
S_j \leq 2 \eps ,\quad j=1, \dots ,m-1.
$$
\elm
\bg \noindent{\it Proof.} \bk
By the formula for $S^{[j]}$ above,
$$
S_j \leq P_j + \frac14 S_{j-1} + \frac14 ( S_{j-1} -P_{j-1} ) = P_j+ 2 \frac14 S_{j-1} \leq P_j+ S_{j-1}/2 .
$$
This implies
$$
S_j \leq \sum_{k=0}^{j-1} 2^{-k} P_{j-k} \leq \eps \sum_{k=0}^{j-1}2^{-k} < 2 \eps.
$$
\hfill {\bg $\Box$}
\section{Formal solution provides analytic with estimates.}\label{s_one_step}
In this section we study a
homological equation \eqref{homol-eq} below with an analytic right-hand side $Q(I,\th)$.
Assuming that it has a formal solution, we will find an analytic one,
and estimate
it in terms of the right hand side. Similar procedures appear in \cite{Ll}.
\blm\label{lemma-homol-eq0}
Let $N_0(I)=I^{tr}\Om I$ where $\Om$ is a symmetric matrix with
$\det\Om\neq 0$, and let $Q(I,\th)$ be analytic in an annulus
$\A_{\rho,\sigma}$ for some $\rho$, $\sigma>0$.
Suppose that the following equation has a formal solution $\widetilde
F (I,\th)$:
\beq\label{homol-eq}
\{N_0,\widetilde F \}= Q.
\eneq
Then equation \eqref{homol-eq} has an analytic solution $F(I,\th)$, defined in
$\A_{\rho,\sigma}$,
and for any $0<\dt<\rho$, $0<\gamma<\sigma$ we have:
$$
|F |_{\rho -\dt, \sigma-\gamma} \leq c(d,\Om) \frac{1}{\dt \gamma^d} | Q |_{\rho, \sigma} ,
$$
where $c(d,\Om)$ is a constant only depending on $d$ and $\Om$.
Moreover, if $Q(I,\th)$ is a homogeneous polynomial in $I$ with coefficients
depending on $\th$, then so is $F(I,\th)$.
\elm
\bg \noindent{\it Proof. } \bk
Expanding $F$ formally into a Fourier series:
$F=\sum_{k\in \Z^d} \widehat F_k(I) e^{2\pi i\< k,\th\>}$,
we get:
$$
\{N_0,F\} = \sum_{j=1}^d F_{\th_j} (N_0)_{I_j} = 2\pi i \sum_{k\in \Z^d} \< k, 2\Om I\> \widehat F_k (I) e^{2\pi i \< k,\th\>}.
$$
Recall that $\Om$ is symmetric, so $\< k, \Om I\>=\< \Om k, I\>$. Expressing $Q=\sum_{k\in \Z^d} \widehat Q_k(I) e^{2\pi i \< k,\th\>}$,
we can rewrite equation (\ref{homol-eq}) as a series of equations indexed by $k$:
\beq\label{homol-eq-k}
\widehat Q_k(I)=4\pi i \< \Om k, I\> \widehat F_k (I).
\eneq
If $\<k, \Om I\> \neq 0$, we can express $\widehat F_k =
\widehat Q_k(I)/ (4\pi i \< \Om k, I\>)$.
Since we have assumed existence of a formal solution of the
homological equation (\ref{homol-eq}) (and hence, a solution of (\ref{homol-eq-k}) for
each $k$), we have:
$$
\< \Om k, I\>=0 \Rightarrow \widehat Q_k(I)=0.
$$
Hence, for $\< \Om k, I\>=0$,
the equation is satisfied for any value of
$\widehat F_k(I)$.
We define $\widehat F_k$ at these points by continuity.
A way to do it is the following.
Differentiate equation (\ref{homol-eq-k})
in the direction of $\Om k$:
$$
\< \Om k, \nabla \widehat Q_k(I) \>= 4\pi i \left( |\Om k |^2 \widehat F_k(I) + \< \Om k, I\>
\< \Om k, \nabla \widehat F_k(I)\> \right) ,
$$
where for a vector $v\in \R^d$ we denote $|v|^2=\sum_{j=1}^d v_j^2$.
For $\< \Om k, I\> =0$, define $\widehat F_k(I)= \< \Om k , \nabla \widehat
Q_k(I)\> /(4\pi i |\Om k|^2)$. Summing up, we have defined a continuous function $\widehat F_k(I) $ by
$$
\widehat F_k (I) = \frac{1}{4\pi i}
\begin{cases}
\< \Om k, I\>^{-1} \widehat Q_k (I), \quad \< \Om k, I\> \neq 0, \\
\frac{1}{|\Om k|^2}\< \Om k , \nabla \widehat Q_k(I)\> ,\ \ \< \Om k, I\> =0.
\end{cases}
$$
Moreover,
since $\widehat F_k(I)$ is analytic in $\D_\rho \setminus \{\< \Om k, I\> =0\}$ and
bounded in $\D_\rho$, it is analytic in $\D_{\rho}$.
Notice that if in equation \eqref{homol-eq-k} $\widehat Q_k(I)$ is a homogeneous polynomial in $I$,
then so is $\widehat F_k(I)$.
Now let us estimate the norm of the solution. Fix $0<\dt<\rho/2$, $0<\gamma<\sigma$. For each fixed
$k\in\Z^d$, we will estimate the corresponding $\widehat F_k(I)$ in two steps: first `$\delta/2$-close" to the resonant plane $\< \Om k, I\> $, and then in the rest of $\D_{\rho -\dt} $.
For the first step, let $\Pi_\dt= \{\<\Om k,I\>=0 \} \cap \D_{\rho-\dt}$ be the part of the
resonant plane falling into $\D_{\rho-\dt}$. Notice that the orthogonal complement to this plane is formed
by the vectors $\alpha e^{2 \pi i \phi } \Om k $, $\alpha\geq 0$, $\phi\in [0,1)$. Let
$$
\Delta=\left\{ I= \alpha \frac{\Om k }{|\Om k | } e^{2 \pi i \phi } \, \Big| \, \alpha < \dt /2, \,\phi\in [0,1) \right\}
$$
be the complex disk
of radius $\dt /2$ centered at zero and orthogonal to $\Pi_\dt$.
Note that the restrictions of
$\widehat Q_k(I)$ and $\widehat F_k(I)$ to this disc
are analytic.
Consider the ${\dt}/2$-neighbourhood $O_\dt$ of
$\Pi_\dt$:
$O_\dt=\bigcup_{I_0\in \Pi_\dt} (I_0+\Delta)$. Then and $O_\dt\subset \D_{\rho -\dt} $.
For each fixed $I\in O_\dt$ there exists $I_0 \in \Pi_\dt$ such that $I\in I_0+\Delta$. We can estimate $|\widehat F_k (I)|$ by
the maximum modulus principle on the disk $I_0+\Delta$. Namely, for $I$ lying on the boundary of this disk we have: $|\< \Om k, I\>|= |\< \Om k, I_0\>+ \< \Om k, \dt \Om k/(2| \Om k|) \> | = | \Om k|\dt/2$. Hence, for such $I$ we have
$$
|\widehat F_k (I)| \leq
\frac{2 | \widehat Q_k |_\rho }{4\pi\dt |\Om k|}< \frac{ | \widehat Q_k |_\rho }{\dt |\Om k|}.
$$
As the second step in this estimate, consider $I\in \D_{\rho -\dt}\setminus O_\dt $. Here
$|\< \Om k, I\>| \geq |\Om k| \dt / 2$, so $|\widehat F_k (I)| $ satisfies the same estimate as above.
By Cauchy estimates, we have:
$$
|\widehat Q_k |_{\rho} \leq | Q |_{\rho,\sigma}e^{-|k|\sigma}.
$$
Since det$\, \Om\neq 0$, there exists a constant $c(\Om)$ such that $|\Om k|\geq |k|/c(\Om)$ for all $k$.
Then
$$
\begin{aligned}
|\widehat F_k |_{\rho-\dt} \leq \frac1{\dt |\Om k|} | \widehat Q_k |_{\rho}
\leq c(\Om) \frac{e^{-\sigma |k|}} {\dt |k|} | Q |_{\rho,\sigma}.
\end{aligned}
$$
Finally, for small $\dt$ and $\gamma$ we have:
$$
\begin{aligned}
|F |_{\rho-\dt,\sigma-\gamma}\leq &
\sum_{k\in \Z^d \setminus \{0\}} e^{(\sigma-\gamma) |k|} |\widehat F_k |_{\rho-\dt }
\leq \frac{c(\Om)}{\dt }
\sum_{k\in\Z^d \setminus \{0\}} \frac{e^{-\gamma |k|} }{|k|} | Q |_{\rho,\sigma}\\
\leq &\frac{c(d,\Om)}{\dt \gamma^d } | Q |_{\rho,\sigma} ,
\end{aligned}
$$
where $c(d,\Om)$ is a constant only depending on $d$ and $\Om$. The estimates above are very wasteful, but they are enough for our purposes.
\hfill {\bg $\Box$}
\section{Proof of Proposition \ref{lemma1}.}
Here we summarize the preparatory work to complete the proof of
Proposition \ref{lemma1}.
Let us return to the original problem.
For a fixed $n$, let the necessary constants be as in Sec.\ref{not_const},
$|\widetilde{R_n} |_{\rho_n } \leq \dt_n^\kappa$,
and
let $ g_{2j}(I)=j\, b_{j} \, (N_0(I))^{j-1}$ as in (\ref{notation_gj}).
\subsection{Estimate of $|\{N_0,F_n \}|_{\rho_n, \rho_n}$ and
$|C_n |_{\rho_n, \rho_n }$. }\label{s_est_NF}
For $j=1, \dots ,m_n-1$ denote
$$
P_{j}:= |N^{[m_n+j]}|_{\rho_0 } +|R^{[m_n+j]} |_{\rho_n } .
$$
By the choice of $\rho_0$, see Sec. \ref{not_const}, for all $j=1, \dots , m_n-1$ we have:
$$
|g_{j}(I) |_{\rho_0} \leq 4^{-j},\quad |N^{[m_n+j]}|_{\rho_0 } \leq \dt_n^\kappa.
$$
Since for $j=1, \dots ,m_n-1$ we have
$|R^{[m_n+j]} |_{\rho_n } \leq |\widetilde{R_n} |_{\rho_n } \leq \dt_n^\kappa$,
and so for these values of $j$ we get
$$
P_j\leq 2\dt_n^\kappa.
$$
let $S_j$ be defined by (\ref{homol-eq-by-order1}). By Lemma \ref{l_est_NF},
for $j=1,\dots m-1$ we have $S_j \leq 2\eps $.
Equations (\ref{homol-eq-by-order_s}) imply that for $j=1,\dots m-1$ we have
\beq\label{eq22}
| \{N_0, F_n^{[m+j-1]} \} |_{\rho_n,\rho_n} \leq S_j\leq 2\eps =4\dt_n^\kappa.
\eneq
By linearity,
$$
| \{N_0, F_n \} |_{\rho_n,\rho_n} \leq \sum_{j=1}^{m_n-1} | \{N_0, F_n^{[m_n+j-1]} \} |_{\rho_n,\rho_n}
\leq 4m_n \dt_n^\kappa \leq 4\dt_n^{\kappa - 1}.
$$
The latter estimate follows from the definition of $m_n$ and $\dt_n$, see Section \ref{not_const}.
Moreover, by \eqref{def_Cn_G},
$$
\begin{aligned}
|C_n |_{\rho_n} &= \sum_{k=1}^{m-2} \left( S_{m-k} \,\sum_{j=k+2}^{m} G_j \right) \leq \sum_{k=1}^{m-2} \left( S_{m-k} \,\sum_{j=k+2}^{\infty} 4^{-j} \right) \\
&\leq \frac{1}{3} \sum_{k=1}^{m-2} 4^{-(k+1)} S_{m-k} \leq \frac{1}{2} \eps=\dt_n^\kappa.
\end{aligned}
$$
Hence,
\beq\label{estimate_Cn_S}
| C_n |_{\rho_n} \leq \dt_n^\kappa .
\eneq
\subsection{Estimates for $F_n$.}
Consider equation \eqref{eq22}.
Lemma \ref{lemma-homol-eq0} with $\rho=\sigma=\rho_n$, $\dt=\gamma=\dt_n$ and $| Q |_{\rho, \sigma} \leq 4\dt_n^\kappa$, implies:
$$
| F_n^{[m+j-1]} |_{\rho_n-\dt_n, \rho_n-\dt_n} \leq 4c(d,\Om) \dt_n^{\kappa - d-1} .
$$
Since $F_n=F_n^{[m_n, m_n+j-1]} $ where $m_n\leq \dt_n^{-1}$, we get:
\begin{equation}\label{eq24}
| F_n |_{\rho_n - \dt_n , \rho_n-\dt_n} \leq \sum_{j=1}^{m_n-1} | F_n^{[m+j-1]} |_{\rho_n-\dt_n , \rho_n-\dt_n} \leq m_n \, 4c(d,\Om) \dt_n^{\kappa - d-1}
\leq \dt_n^{\kappa - d-3} \leq \dt_n^{3}.
\end{equation}
The latter estimate follows from the definition of $\kappa$, see Section \ref{not_const}.
\subsection{Estimates for $\Phi_n$.}
Here we prove that with $F_n$ as above, estimates (\ref{estPhi}) and (\ref{estPhi-1})
hold true. Indeed,
the coordinate change $\Phi_n = X_{F_n}^1$ is the time one map of the flow $X_{F_n}^t$
defined by the equations
$$
\dot I = \partial_\th F_n (I,\th), \quad \dot \th =-\partial_I F_n (I,\th).
$$
By \eqref{eq24} and Cauchy estimates we get
\begin{equation}\label{eq_est_dF}
| \partial_I F_n |_{\rho_n - 2\dt_n , \rho_n-\dt_n} \leq \dt_n^{2}, \quad | \partial_\th F_n |_{\rho_n - \dt_n , \rho_n-2\dt_n} \leq \dt_n^{2}.
\end{equation}
Then for any $t\leq 1$:
$$
|X_{F_n}^t (I,\th)- (I,\th)|_{\rho_n - 3\dt_n , \rho_n- 3 \dt_n} \leq t \, \dt_n^{-1} | F_n |_{\rho_n - 2\dt_n , \rho_n-2\dt_n} \leq \dt_n^{2}.
$$
\begin{equation}\label{eq_est_Phi}
X_{F_n}^t :\A_{\rho_n-3\dt_n,\rho_n-3\dt_n} \mapsto \A_{\rho_n-2\dt_n,\rho_n-2\dt_n}
\end{equation}
In particular, since $ \Phi_n = X_{F_n}^1$, we get the desired formulas \eqref{estPhi} and \eqref{estPhi-1}.
\subsection{Estimate of the new remainder $\widetilde {R_{n+1}}$.}
\blm
\label{lemma_est_Rn+1} For $F_n$ constructed above, estimate
(\ref{estRn}) holds:
$$
|\widetilde {R_{n+1}} |_{\rho_n - 3\dt_n , \rho_n-3\dt_n} < 4 \dt_{n}^\kappa.
$$
\elm
\bg \noindent{\it Proof.} \bk
By Lemma \ref{lem_homol_eq},
$$
\widetilde {R_{n+1}}= A_n+B_n+C_n,
$$
where $A_n$, $B_n$ and $C_n$ are defined by
(\ref{not_AnBn}) and (\ref{not_Cn}).
{\bf Estimate of $A_n$: }
Using \eqref{eq_est_Phi},
we get:
$$
| \widetilde {R_{n}}^{[>m_{n+1}]}\circ \Phi_n |_{\rho_n-3\dt_n,\rho_n-3\dt_n}
\leq |\widetilde {R_{n}} |_{\rho_n-2\rho_n,\rho_n-2\dt_n} \leq \dt_n^\kappa.
$$
\medskip
{\bf Estimate of $C_n$:} We showed in section \ref{s_est_NF} that
$$
|C_n|_{\rho_n,\rho_n}\leq \dt_{n}^\kappa.
$$
\medskip
{\bf Estimate of $B_n$:}
By \eqref{eq_est_dF},
$
| \partial_I F_n |_{\rho_n - 2\dt_n , \rho_n-\dt_n} \leq \dt_n^{2}$ and $ | \partial_\th F_n |_{\rho_n - \dt_n , \rho_n-2\dt_n} \leq \dt_n^{2}.
$
By (\ref{estRn})
$$
|R_{n} |_{\rho_n,\rho_n} \leq |\widetilde {R_{n}} |_{\rho_n, \rho_n} \leq \dt_n^\kappa.
$$
This implies, using Cauchy estimates, that
$$
| \{ R_n,F_n\} |_{\rho_n - 2\dt_n , \rho_n-2\dt_n} \leq \dt_n^\kappa.
$$
Notice that, by formulas \eqref{eq_homol} and \eqref{not_Cn}, we have
$ \{ N_n, F_n \} = R_{n} +N_{n} - N_{n-1} +C_n$.
By \eqref{est_N},
$$
|N_{n} - N_{n-1}|_{\rho_0,\rho_0} =\sum_{j=1}^{m_n}N^{[m_n+j]}\leq m_n \dt_n^{\kappa+1}\leq \dt_n^{\kappa}.
$$
and therefore
$$
| \{ N_n, F_n \} |_{\rho_n,\rho_n} = |R_{n} |_{\rho_n,\rho_n} + |N_{n} - N_{n-1} |_{\rho_n,\rho_n}+|C_n|_{\rho_n,\rho_n}
\leq 3 \dt_{n}^\kappa.
$$
Combining the above estimates, we get
$$
| \{ \{ N_n, F_n \} ,F_n\} |_{\rho_n - 2\dt_n , \rho_n-2\dt_n} \leq \dt_n^\kappa,
$$
Since, by \eqref{eq_est_Phi}, for any $t\leq 1$ we have $X_{F_n}^t :\A_{\rho_n-3\dt_n,\rho_n-3\dt_n} \mapsto \A_{\rho_n-2\dt_n,\rho_n-2\dt_n} $,
we obtain
$$
| \{ \{ N_n, F_n \} +R_n,F_n\} \circ X_{F_n}^t |_{\rho_n-3\dt_n,\rho_n-3\dt_n} \leq
| \{ \{ N_n, F_n \} +R_n,F_n\} |_{\rho_n-2\dt_n,\rho_n-2\dt_n} \leq 2 \dt_n^\kappa.
$$
\hfill {\bg $\Box$}
\bigskip
Here we get the desired estimate for the remainder term. We have proved above that
$$
|\widetilde {R_{n+1}} |_{\rho_n-3\dt_n,\rho_n-3\dt_n}< 4 \dt_{n}^{\kp}
$$
Recall that
$\widetilde {R_{n+1}}=\widetilde {R_{n+1}}^{[>m_{n+1}]}$.
By Lemma \ref{lemma_est_Rn+1_final} proved below, this implies the desired estimate
$$
|\widetilde {R_{n+1}} |_{\rho_{n+1},\rho_{n+1}}< \dt_{n+1}^{\kp}
$$
This finishes the proof of Proposition \ref{lemma1}, and hence Theorem 1 (as explained
in the introduction). \hfill {\bg $\Box$}
\medskip
\blm
\label{lemma_est_Rn+1_final}
Suppose that the constants $\kp$, $b$, $\dt_n$, $q_n$, $\rho_n$ are defined in Section \ref{not_const}, and
an analytic function $G(I,\theta)$ satisfies $G=G^{[>m_{n+1}]}$, and
$$
|G |_{\rho_n-3\dt_n,\rho_n-3\dt_n}< 4 \dt_{n}^{\kp}.
$$
Then
$$
|G|_{\rho_{n+1},\rho_{n+1}}< \dt_{n+1}^{\kp}.
$$
\elm
\bg \noindent{\it Proof. } \bk
By the definition of $\kp$ in Section \ref{not_const} we have:
$q_n^{m_{n+1}+1}=q_{n}^{2^{n+1}+2} < q_{n}^{2^{n+1}} = 2b= 2^{-\kp-2}$.
Also recall that $\dt_{n+1}=2^{-1}\dt_{n}$.
Since $G$ starts with terms of degree
$m_{n+1}=2^{n+1}+2$, we have:
$$
|G |_{q_n(\rho_n-3\dt_n),q_n(\rho_n-3\dt_n)}< q_n^{2^{n+1}+2} \,4 \dt_{n}^{\kp}
\leq 2^{-\kp-2} \, 4 \dt_n^{\kp} \leq \dt_{n+1}^{\kp}.
$$
\hfill {\bg $\Box$}
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|
TITLE: Decompose and compute the sign of $\sigma(k)=n+1-k$
QUESTION [0 upvotes]: Let $n\geq 2$ and $\sigma$ is permutationof $\{1,2,\ldots,n \}$ defined by :
$$\sigma(k)=n+1-k$$
Decompose permutation $\sigma$ into product of disjoint transpositions and compute the sign of it ?
indeed,
Here is solution from book but i can't understand it would someone elaborate it please
$$\forall k\in \{1,2,\ldots,n \}\ \sigma(k)=n+1-k \mbox{ and } \sigma(n+1-k)=k$$
So there are two cases according to the parity of n.
If $n$ is even: $n=2m$, where $m \in \mathbb{N}^*$. Then
$$\prod_{k=1}^m\left(k,2m+1-k \right) \mbox{ and } \operatorname{sign}(\sigma)=(-1)^{m} $$
If $n$ is an odd number, $n=2m+1$ with $m\in \mathbb{N}^*$. So
$$\prod_{k=1}^m\left(k,2m+2-k \right) \mbox{ and } \operatorname{sign}(\sigma)=(-1)^{m} $$
REPLY [0 votes]: let $n=2m $ or $n=2m+1$ according to $n$ even or odd,
let $\gamma = \Pi_{1\leq i\leq m} \tau_{i,n + 1-i}$ as product of
m disjoint transposition, then we check
$\gamma(k)=n+1-k=\sigma(k) \forall k\in \{1,\cdot\cdot\cdot,n\}$.
it may be noted that if $n=2m+1$, then m+1 is a fixed point ie
$\gamma(m+1)=m+1=\sigma(m+1)$, also $sign(\sigma)=(-1)^m$.
| 124,291
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1-866-309-1962
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2014 GMC Savana G33903 Light and Medium Duty Trucks
Unit #9168375 / Light Duty Box Truck
BAKERSFIELD, CA
$ 22,750 80,15533903
- Color
- YELLOW
- A/C
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- GVW
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- Wheel Base
- 177
Engine
- Manufacturer
- GMC
- Model
- 6.0LV8
- Horsepower
- 342
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Drive Axle
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- 1
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- 8600
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- Hydraulic
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- LT22575R16
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- Radial
Transmission
- Type
- Automatic
- Manufacturer
- GMC
- Model
- 6L90E
- Speeds
- 6
Fuel
- Fuel Tanks
- Single
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- 33
Body
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- Supreme
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- 16'0"
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| 19,910
|
Resource type:
Videos —
Date posted:
January 18th, 2018
Number of views:
510
BBS Sport launch a new active website promoting movement that can be done in Class and it aims to support learning concepts such as Times Tables
SERC / Newbridge School
Barley Lane Campus, 258 Barley Lane
Goodmayes
Ilford, IG3 8XS
SERC is a service and information centre brought to you by Redbridge.
| 282,407
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It is against the law for someone to intrude on your privacy in this way and offenders can be prosecuted under the Computer Misuse Act.
Along with your email account, you also can use your Hotmail email address to sign into Windows Live Messenger, which allows you to connect with your contacts in Hotmail sending instant messages, photos and files, and you can even video chat with your webcam.
It’s also important to consider some of the technical risks and to help prevent these by making sure you think about where you webcam is, what sites and services you use and what steps you can take to protect your webcam.
Webcams make it possible for us to chat face to face with friends and family wherever we, or they, may be.
This can then be recorded by the watcher and be potentially used to threaten or blackmail.
Many of these ' Rats' now include a function allowing a hacker to access the victim's webcam without their knowledge.
So how does a hacker get a virus into your computer or device?
There are a huge number of positive uses and potential for using webcams as a tool for communication.
Our advice can help you minimise the risks associated with webcams and video chatting.
| 70,132
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The Lowell Sun has a pretty big article today on Dracut elected officials who are on the town's health insurance program. Former Selectman Warren Shaw, who served on the board for over twenty years, and Kathleen DiTillio, widow of former Selectmen Jack DiTillio, both receive health insurance through the town. Both Shaw and DiTillio say they are eligible for the insurance and DiTillio does pay $3,800 a year in premiums. Shaw of course operates Shaw Farm whereas DiTillio moved to Florida after her husband passed away in 2005.
The article quotes Roger Daigle who many know as one of the talking heads on "Inside Dracut Politics" who says, "They're all bloodsuckers in my book. It galls me how these people make up their own rules. The law says they may be eligible, not that they are eligible. They're supposed to work 20 hours a week and the benefits are supposed to end when their term of office expires. No elected official works 20 hours a week. They pick and choose what part of the law suits them."
Calling Shaw and DiTillio bloodsuckers is over the top but anyone who has seen his show and read his comments on a certain blog know Daigle has made a career on being caustic and inflammatory.
In addition to Shaw and DiTillio, the article points out that four of the five selectmen are on the town's insurance. The only one who doesn't is Jim O'Loughlin who is a state employee so he gets it through them. Joe DiRocco gets it as a retired fire fighter, John Zimini through his wife, a school employee, and Robert Cox and George Malliaros have opted for the insurance (Cox owns Coyle's Tavern and Malliaros is a local attorney).
It is an interesting story, you can see it all here.
Friday, November 14, 2008
Health Benefits for Selectmen
| 321,960
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TITLE: Why isn't the path integral defined for non-homotopic paths?
QUESTION [16 upvotes]: Context
In the Aharonov Bohm effect, there is a solenoid which creates a magnetic field. Since the electron cannot be inside the solenoid, the configuration space is not simply connected.
Question
I've read in this paper, that the path integral is defined only for paths in the same homotopy class in the configuration space. But I don't see the reason for this. Could someone explain it or give any reference?
It seems that Laidlaw, DeWitt and Schulman have done some work, but I didn't see any proof. And Feynman & Hibbs don't seem to mention it.
Furthermore, does the same problem arise in standard variational calculus when one applies Hamilton's principle?
REPLY [3 votes]: TL;DR: The Feynman path integral/kernel/amplitude $K(x_f,x_i)$ is in general a weighted sum over ALL (not necessarily homotopic) paths, as user Heidar writes in a comment above.
In more detail: Let there be given an initial point $x_i$, a final point $x_f$, and a fiducial point $\ast$. Fix two paths $\gamma_i: x_i\to \ast$ and $\gamma_f:\ast \to x_i$. It is natural to assume that the full path integral is of the form $$K(x_f,x_i) ~=~ \sum_{\gamma\in \pi^1(X, \ast)} \chi(\gamma)~ K^{\gamma}(x_f,x_i),$$ where $\chi(\gamma)\in\mathbb{C}$ is some weight. Here the partial path integral $K^{\gamma}(x_f,x_i)$ consists of all paths $x_i\to x_f$ in the homotopy class $[\gamma_i+\gamma+\gamma_f]$. In this way we have formally counted each path exactly once.
For consistency, it turns out that $\chi$ must be a 1-dimensional unitary representation of the fundamental group $\pi^1(X, \ast)$ using well-known group and probability properties of the path integral, cf. the theorem of Ref. 1. The upshot is that we have reduced the full path integral $K(x_f,x_i)$ to a weighted sum of partial path integrals $K^{\gamma}(x_f,x_i)$ whose paths are all within the same homotopy class. But we still have to define/calculate the weighted sum $K^{\gamma}(x_f,x_i)$ of paths within each homotopy class.
References:
M.G.G. Laidlaw & C.M. DeWitt,
Phys. Rev. D3 (1971) 1375.
| 127,024
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I’ve heard so many things about escape rooms and even seen them featured on TV multiple times but have never had the opportunity to go to one myself. For the first time last week I was able to experience an escape room!
Since they were hosting some of us DFW Bloggers ,they served us refreshments before sending us off into the game.
Fun Fact: Only 25{7f5ea8dd8d8b02f00b4c8670c4d90eb669b4be3027fc27eb10848957f8ea2e12} of people escape the room within their allotted 60 minutes!
After hearing that I instantly assumed, well this is it, I’m going to be locked into a room for 60 minutes and won’t get out until time runs out. It took six of us, with everyone participating, to be able to solve all of the puzzles. I can’t imagine only two people doing it, which is the minimum for the smaller rooms.
We solved all clues and riddles with only 13 minutes to spare! We get to say we are part of the 25{7f5ea8dd8d8b02f00b4c8670c4d90eb669b4be3027fc27eb10848957f8ea2e12}. I will say this Escape Hunt Dallas is definitely not what I expected it to be. You know you build up in your mind what something is going to be like and the experience that we had was so much better than I could have imagined.
Our group participated in the Texas Loan Star room. Which has a max occupancy of 6 players, since it’s the smallest game they have.
Currently, they have three different games available with unique themed rooms. The rotation is roughly 18 months and then it’s time for fresh new challenges.
Texas Loan
When walking into the building you automatically feel like you are in a mystery. They have old photos hanging on the walls and pipes sitting on end tables.
To keep the mystery alive and prevent people from sharing the puzzles, they have lockers for you to keep your stuff in while you are in the game. The recommended age is 10-years-old, based on the level of difficulty figuring out the clues. Children must be accompanied by an adult in the escape room.
| 99,323
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April 9, 2018 Angie & Chris | Colorful Old Town Alexandria Wedding First day happen! Ceremony Venue | The Alexandrian Hotel Reception Venue | Virtue Feed and Grain Hair & Makeup | Modern Bridal Flowers | Helen Oliva Flowers Dress | BHLDN Band | Mix Tape « Bachelorette Party Ideas | #MarriedMedlinsPaula & Jordan | Manassas Battlefield Engagement Session »share to: no comments Your email is never published or shared. Required fields are marked * Name * Email * Website Comment Notify me of follow-up comments by email. Notify me of new posts by email.
| 103,710
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Services /
Calculators / Newsletter /
Archives
Ohio Sales Tax Changes
At present, sales of licensed motor
vehicles, boats and outboard motors are subject to the sales tax rate of
the residence of the purchaser and the tax goes to that county.
All other taxable sales are subject to the sale tax rate of the location
of the vendor's place of business and the tax goes to his county.
Starting January 1, 2004, there will be
major changes to which rate and county will be used.
Sales of licensed automobiles,
boats, and outboard motors will continue to use the rate of the county
residence of the purchaser. Single payment leases will
also be treated this way.
Leases that require periodic payments
of other property will use the rate and the county where the leased
item is first used.
For all other sales, the
following rules apply:
If the customer receives the
merchandise or service at the vendor's place of business, the rate and
county of the place of business is to be used.
If the customer does not receive the
merchandise or service at the vendor's place of business, the rate and
county where the customer receives the merchandise or service is
to be used.
If the location that the customer
receives the merchandise or service is unknown, the rate and the county
of the purchaser obtained from the vendor's business records is to be
used as long as this does not constitute bad faith.
These changes will require most
businesses to make major changes to their accounting systems to record
taxable sales by the county of place of business for merchandise and
services received at that location and by the county in which the
merchandise and services are delivered. Businesses will also be
required to keep track of the sales tax rate of all 88 counties of Ohio
and how much tax is collected for each county each month. Contact
our office if you need help with this area.
Consumers will be able to minimize sales
tax by some simple steps.
If you shop in a lower sales tax rate
county, take delivery at the store in which you shop.
If you shop in a higher sales tax rate
county, have the store deliver the merchandise to your home or business,
which ever has the lowest rate.
If you lease items other than motor
vehicles, boats, and outboard motors, try to use them for the first time
in a county with the lowest sales tax rate.
We will be pleased to discuss these
issues with you and plan to reduce your sales tax liability to the minimum
amount due.
(November, 2003 Newsletter)
James E. Newland, CPA
939 Center Road
Eastlake, Ohio 44095
440-951-9799
Service@NewlandCPA.com
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'Trojan Women' To Explore War Aftermath With 'Alluring Imagery'
Oct. 11, 2013
SHSU Media Contact: Emily Schulze Binetti
Sam Houston State University’s Department of Theatre and Musical Theatre will present Euripides’ Trojan Women Wednesday (Oct. 16) through Friday (Oct. 19) in the University Theatre Center’s Erica Starr Theatre.
Performances will begin at 8 p.m. each evening, with a 2 p.m. Saturday matinee.
Adapted from Gilbert Murray’s rhyming verse translation, the well-known Greek classic explores the timeless meditation on the moments of individual choice that separate death and life, despair and hope, future and past.
Taking place before the walls of Troy, an ancient city near the western coast of present-day Turkey, the story unfolds at dawn, a day after Greek armies won the Trojan War.
The Trojan women—including Hecuba, the queen of Troy—congregate outside the walls of the burning city in deep despair. Still mourning the slaughter of their husbands and sons, the women await enslavement and exile.
Despite the tragedy of the story, SHSU’s production is designed to give the audience a visually alluring experience.
“My directorial concept is driven by the need to transform a horrific subject into something aesthetic pleasing, even beautiful,” said the show’s director and professor of theatre David McTier.
To create the visual concept, McTier recruited SHSU graduate dance student Travis Prokop to provide original choreography for three modern dances in the production.
Additional creative talents were provided by senior theatre majors Peter Ton for video production and Colton Spurlock for scenic design, described by McTier as “very sculptural.”
In trying to connect new audiences with the story of Trojan Women, the costumes were inspired by the modern fashions of designer Alexander McQueen.
“I designed the show using, among other things, the book on McQueen's work ‘Savage Beauty’ combined with historical research of Greek armor and fashion to twist the garments into a new created period,” said costume designer and theatre faculty member Kris Hanssen. “This new period combines modern fit and construction along with an organic decomposition played out with the silhouette of the ancient Greek fashion.”
The Trojan Women cast includes Katelyn Johnson, Sara Myers, Latoya Curtis, Samira Williams, Adrienne Whitaker, Nathan Wilson, Thomas Williams, Sean Willard, Adriana Dominguez, Maggie Ellison, Tiffany Figueroa, Monique Lott, Monae Lott, Monica Malone, Laura Norby, Kiara Steelhammer, Adrienne Whitaker, Courtney Williams, Audrey Wilson, Chelsea Womack, Trey Brake, Joe Daniels, Blake Jackson, Bryan Okoro and Sean Willard.
Tickets are $15 for general admission and can be purchased at the UTC Box Office, by phone at 936.294.1339, or online at shsu.edu/academics/theatre/tickets.
-.
| 244,475
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TITLE: Square integrable borel probability measures on Euclidean spaces are the law of random variables from an atomless polish space
QUESTION [0 upvotes]: Could someone provide me with a reference or proof for the following: Let $(\Omega, \mathcal{A}, P)$ be an atomless probability space, with $\Omega$ a Polish space. Given $f$ a random vector on $(\Omega,\mathcal{A},P)$ denote by $L(f)$ its law, i.e. the induced measure on the Euclidean target space. Then, for any $k$, and any square integrable measure $\mu$ on $\mathbb{R}^{k}$, there exists a random vector $f$ on $(\Omega,\mathcal{A},P)$ with $L(f)=\mu$.
In Cardaliaguet's Notes on Mean Field Games, subsection 6.1 pg 43, he mentions the result as 'recall'. Meanwhile Keisler, Sun on Why saturated probability spaces are necessary, Lemma 2.1(ii), they mention something stronger as a well-known result.
REPLY [1 votes]: The proof I like to remember goes as follows.
Since all uncountable Polish spaces are Borel isomorphic, it suffices to prove this for $\Omega = \mathbb{R}$ and $k=1$, so that $P, \mu$ are both measures on $\mathbb{R}$, with $P$ atomless. Let $F,G : \mathbb{R} \to [0,1]$ be their respective cumulative distribution functions. Then the pushforward $F_* P$ is Lebesgue measure $\lambda$ on $[0,1]$ (easy exercise; show that $(F_* P)([0,a]) = a$). Now let $G^\leftarrow : [0,1] \to \mathbb{R}$ defined by $G^\leftarrow(t) = \sup\{ x : G(x) < t\}$ be the "inverse" of $G$. Show that $G^\leftarrow_* \lambda = \mu$ (also an exercise, or see Theorem 1.2.2 of Durrett, Probability: Theory and Examples). We conclude that $(G^{\leftarrow} \circ F)_* P = \mu$, which is to say that if we consider $G^{\leftarrow} \circ F$ as a random variable on $(\mathbb{R}, P)$, its law is $\mu$.
The square integrability is not needed, and this works equally well if $\Omega$ and $\mathbb{R}^k$ are replaced by any other Polish spaces.
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West Bloomfield
If you are moving to West Bloomfield or South Eastern Michigan then you should look at the Oakland County area. It is only a 30 -45 minute ride to downtown and shorter to Chrysler, or Ford headquarters. West Bloomfield is a beautiful community with rolling hills and well designed communities. West Bloomfield is an area of plentiful entertainment, filled with a variety of restaurants, shopping and the best recreational facilities in Oakland County, Michigan, including a private country club on the shore of Orchard Lake. As of 2008, West Bloomfield Township was ranked the 8th highest income city in the United States with a population of at least 50,000.
The township is a multi-cultural city with many different cultures that have blended together to make a great city. You can get almost any type of ethnic food and there are so many different types of place to worship.
West Bloomfield is what I would call a "safe" community. The media plays up "Detroit crimes and Detroit's problems" but that is totally different from West Bloomfield. You can safely walk the streets of West Bloomfield at night. You don't have to worry about getting shot, or having your car stolen. Sure there is random crime but you can safely go the store, or go out to eat at any time of the day or night. For more detailed info on West Bloomfield crime statistics google "West Bloomfield crime statistics"
Find your homes value instantly
West Bloomfield Real Estate
West Bloomfield real estate is mainly colonial homes. Most of the homes built after the 1970's were colonials. I'm not saying there are not farm houses, bungalows, ranches, or split level homes. There are, but not in the quanity of colonial style homes.
West Bloomfield real estate ranges in age from the 1920 to homes that were built this year. The size of homes range from 700 square foot lake cottages to eleborate estates over 9000 square feet in size. There is every shape and style of real estate for you to look at.
There are quite a few West Bloomfield condominium complexes. Many of the condos are built closer to the main road. West Bloomfield condos range from 700 square feet to over 3000 square foot. Those most of them range between 1000-2000 square feet. When you list your home it pays to stage your West Bloomfield home
Click this link to learn more about West Bloomfield foreclosures
West Bloomfield Lake homes
West Bloomfield Township is sometimes referred to as the "lake township of Oakland County being heavily dotted with small pond sized lakes and medium-sized lakes. the township has a total area of 31.2 square miles of which 3.9 square miles of the township (12.49%) is water. West Bloomfield has 24 lakes that cover 1858.7 acres. So there are plenty of West Bloomfield lake front homes to chose from.
West Bloomfield Lake front real estate varies from small 1920 lake cottages to large luxury lake front mansions costing several million dollars. West Bloomfield lake front properties command a higher price that surrounding cities lake properties because of the school system. Many of the old lakefront cottages have been torn down and re-built. So many of West Bloomfield lakes have a higher concentration of newer and bigger lake front homes.
Here is a list of West Bloomfield Lakes. Bloomfield Lake, Brookfield Lake, Cass Lake, Cross Lake, Darb Lake, Flanders Lake, Fox Lake, Green Lake, Hammond Lake, Lake Marion, Middle Straits Lake, Mirror Lake, Moon Lake, Morris Lake, Orchard Lake, Pine Lake, Pleasant Lake, Rockwell Lake, Scotch Lake, Simpson Lake, Union Lake, Upper Straits Lake, Walnut Lake, Woodpecker Lake. Some of these lakes do not entirely lie in West Bloomfield. Some are only partially in the township.
West Bloomfield Demographics
Based on the 2000 census data, median age for residents in West Bloomfield Township, MI is 42.4 (this is older than average age in the US.
The average household size was 2.74 and the average family size was 3.17.
West Bloomfield is known for its large Jewish populations. It is home to the Jewish Community Center of Metropolitan Detroit, several Jewish schools such as the Jewish Academy of Metropolitan Detroit and the Frankel Jewish Academy, and to the museum of the Jewish Historical Society of Michigan.
West Bloomfield Schools
Most of the cities children go to West Bloomfield School system but there are a few areas where the children do go to Walled Lake or Pontiac Schools
West Bloomfield Library
The West Bloomfield Township Public Library has been named one of 10 recipients of the 2010 National Medal for Museum and Library Service, the nation’s highest honor for museums and libraries. The annual award, made by the Institute of Museum and Library Services (IMLS) since 1994, recognizes institutions (5 libraries and 5 museums) for outstanding social, educational, environmental, or economic contributions to their communities
West Bloomfield Weather
The average snowfall in West Bloomfield is about 40 inches a year. So many people when moving to West Bloomfield or Southeastern Michigan are worried about the winters. Sure we get snow, but it is not overwhelming that you have to stay inside all the time. West Bloomfield Township doesn't have much in the way of natural disasters. No Hurricanes, little flooding, and only one tornado in the last 35 years. The township was struck by an ice storm on March 2, 1976 and again on March 14, 1997 (the worst in state history). A tornado struck on March 20, 1976, killing one person, damaging and/or destroying ninety five residences. This tornado was largely responsible for the initiation of the tornado warning system in Oakland County.
West Bloomfield Lakes
The major lakes in West Bloomfield are:
Bloomfield Lake 5.6 acres
Brookfield Lake 4 to 5 acres
Cass Lake 1,280 acres, 123 ft. max. depth largest lake in Oakland County Public access
Cross Lake 7.3 acres
Darb Lake 16.5 acres
Flanders Lake 9.6 acres
Fox Lake 6.2 acres
Green Lake 166 acres, 65 ft. depth electric motors only
Hammond Lake 85 acres
Lake Marion 6 acres
Middle Straits Lake 171 acres, 55 ft. depth
Mirror Lake 11 acres, 25 ft. depth
Moon Lake 3.7 acres
Morris Lake 15 acres
Orchard Lake 788 acres, 110ft. depth Public Access
Pine Lake 395 acres, 90 ft. depth
Pleasant Lake 41
Rockwell Lake 2.7 acres
Scotch Lake 9.3 acres
Simpson Lake 13.2 acres
Union Lake 465 acres, 102 ft. depth
Upper Straits Lake 323 acres, 96 ft. depth
Walnut Lake 232 acres, 101 ft. depth
West Bloomfield Lake 12 acres
Woodpecker Lake 16 acres
West Bloomfield Recreation
West Bloomfield Parks and Recreation Maintains 516 acres of parkland in addition to using a portion of the 99 acre civic center and the new recreation community center at the civic center campus. These are: Bloomer Park, 7581 Richardson Road, is a 36 acre park on Middle Straits Lake,featuring a picnic area, grills, picnic shelter, nature trails, large accessible playscape, boat launch, cross country skiing, restrooms, drinking water, and a storage facility.
Some of the other parks are Bloomfield Knolls, Civic Center Park, West Bloomfield Community Sports Park, Drake Community Sports Park, Karner Farm, Marshbank Park, Schulak Farm, Sylvan Manor Park, West Bloomfield Woods Nature Preserve, West Bloomfield Trail Network, West Bloomfield Recreation Community Center,The West Bloomfield Family Aquatic Center
These parks have everything from baseball diamonds, tennis courts, nature trails, off lease dog areas, and much more. Click on the park link above to get more indepth information
Click here for information on Top West Bloomfield real estate agents
So you are thinking of moving to West Bloomfield and you are wondering how safe West Bloomfield is and how safe the communities around are. As a local West Bloomfield realtor I can tell you that West Bloomfield is a very safe community. They say West Bloomfield is safer than the majority of cities, towns, and villages in America (77%) and also has a lower crime rate than 78% of the communities in Michigan.
West Bloomfield Michigan has no:
West Bloomfield has a large and good police force. You are not going to get carjacked going into a gas station in West Bloomfield. You are not going to get mugged or robbed when going for a walk in your neighborhood. 72% of the people say there is a strong police presence in the township.
One website gave the city an A+ grade. The A+ grade means the rate of crime is much lower than the average US city. West Bloomfield is in the 93rd percentile for safety, meaning 7% of cities are safer and 93% of cities are more dangerous.
You have a one in 1491 chance of being a victim of violent crime in West Bloomfield, whereas the averages in Michigan is one in 229.
Whether you are going shopping or going to a park you won't have the fear of getting robbed or molested. Things that you see happening on the news in Central Park in New York is not going to happen in West Bloomfield. Yes, occassionally you have home break ins but you don't have home invasions.
How To Stage Your West Bloomfield Home To Sell
West Bloomfiled Foreclosures
West Bloomfield Home Inspectors
Watch our Big Lake Video
Learn more about Judah Lake & Big Silver Lake
Search Clark Lake Livingston County Michigan
#westbloomfieldmi #westbloomfieldhomesforsale #westbloomfieldrealestate #homesforsalewestbloomfieldmi #westbloomfieldcondosforsale #westbloomfieldtwpmi #westbloomfieldtownshipmi #Westbloomfieldcondominium #westbloomfieldforeclosures #howsafeiswestbloomfield #Westbloomfieldcrimestatics #westbloomfieldschools
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As a Search Engine, Bing is next to Google. And it's now concerned to increase the market share. Search engine Bing are now offering special features to the users. In this post, I'm gonna talk about different uses of Bing.
Bing can be used for multiple purposes. But I would like to concentrate on 4 amazing features of Bing. You can use Bing as -
Bing can be used for multiple purposes. But I would like to concentrate on 4 amazing features of Bing. You can use Bing as -
- Calculator - Perform basic and some scientific calculations.
- Unit Converter - Perform unit conversions.
- Clock - Check the time of different cities with date.
- Dictionary - Look up the word meaning.
Note: One or more of the above features may require Internet Explorer to function. Some of these features may not be available in some countries.
Recommendation
I would like to recommend you to use Internet Explorer when you're trying to use these features. And make sure you're on. Let's go -
Bing as Calculator
Or simply type 45 - 9 and hit Enter. The result will be shown on Bing Calculator.
Bing as Unit Converter
- Type - Unit converter or Convert unit and press Enter
- Choose your desired unit from the drop down box.
- Select the format that you need to convert from the left, and the format in which you want your result from the right.
- Enter the value and get the result.
Or you can simply type, 35 Celsius in Fahrenheit and Bing will show automatically show the result in its Unit Converter.
Bing as Clock
You may have watch, smart phone or anything which is enough to see the time. What, if you need to see the time of another country? With Bing, you can see the date and time of any city/ country of the world within a second!
- Type anything as - Dhaka Time/ Singapore Time/ Sydney Time/ London Time
- Immediately, you will get the Current Time of that city/ country with date
Bing as Dictionary
- Type anything as - Define Equilibrium/ GDP/ Diffusion
- The meaning of the word will be right there with synonyms, explanation and pronunciation!
MSN Weather Forecast on Bing Search Result
Bing also shows weather forecast data from MSN. Suppose, you need to know the weather forecast of a city. Simply type as -
Dhaka Weather or London Weather or any city. The result will be shown as follow -
Undoubtedly, Google is still playing the role of leader in search market. And Bing is also playing as an active threat there. As new features are being added to Bing, we expect much severe competition in the future.
By the way, you may also read - Different Search Techniques of Google . . .
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You Got It (The Right Stuff)
"You Got It (The Right Stuff)" is a 1988 single from New Kids on the Block. The second single...
File history
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Testing Service (NTS) only for females belongs to KPK and FATA.
Schools Details for 04 Years General Nursing Diploma: 2016-20:
- School of Nursing Khyber Teaching Hospital Peshawar
- School of Nursing Lady Reading Hospital Peshawar
- School of Nursing Saidu Sharif Swat
- School of Nursing Mardan Medical Complex Mardan
- School of Nursing Ayub Teaching Hospital Abbottabad
- School of Nursing Hayatabad Medical Complex Peshawar
- School of Nursing Liaquat Memorial Hospital Kohat
- School of Nursing Bannu
- School of Nursing Mufti Mahmood Memorial Hospital D.I Khan
- School of Nursing District Headquarter Hospital Abbottabad
Schools Details for 02 Years Lady Health Visitor (LHV) Course: 2016-18:
- Public Health School Nishtarabad Peshawar for following districts:
Peshawar, Nowshehra, Charsadda, Sawabi, Mardan, Kark, Kurram, Karam Agency, Bonir, Shangla, Chitral, Khyber Agency, Orakzai Agency & North Waziristan, Agha Khan Health Services Chitral, AJK & Self Finance Status.
- Public Health School Hayatabad Peshawar for following districts:
Swat, Malakand, Dir Upper, Dir Lower, Kohat, Hangu, Mehmand Agency, Bajor Agency, F.R Peshawar, F.R Kohat and Self Finance.
- Public Health School Abbottabad for following districts:
Abbottabad, Mansehra, Haripur, Batgram, Kohistan, AJK & GB.
- Public Health School Dera Ismail Khan for following districts:
D.I Khan, Bannu, Tank, South Waziristan Agency, F.R Lukcy, F.R Bannu, F.R Lucky Marwat, F.R D.I Khan, F.R Tank and Self Finance.
Qualification:
Matriculation or F.Sc (Pre Medical) or equal.
General Information:
Nursing Students will entertain with 500/- Rs and Selected candidates for LHV course on regular basis will be entertained with 3565/- Stipend. Candidates must attach Matric/F.Sc certificates and DMC, Domicile, Computer CNIC/Form B and 2 recent photographs with application forms and send it to National Testing Service (NTS) address as mentioned into below advertisement. Complete details are also available at National Testing Service (NTS) website: Click Here
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Your Now view shows you what matters most to you in the moment - the right information for wherever you are or need to be. Here, you can quickly see the date and time, see what’s next in your calendar, or change your display and volume settings. Think of it like your home screen for Focals.
Now moments
Contextual, helpful, but never overbearing, Now moments help you get the information you need exactly when and where you need it. You won’t see all Now moments at the same time, but here are a few of our favourites you could see on your display:
- See your shopping list appear after you’ve entered a grocery store
- Discover new restaurants and coffee shops around you. Get directions and reviews in your line of sight so you can travel with confidence
- See what’s currently playing on a Spotify playlist
Settings
It’s easy to customize how you see and interact with what pops up on your Focals display. Move up on your Now view to see your brightness, do not disturb, and volume settings on Focals. Click each icon to cycle through your settings:
- 3 brightness settings
- 4 volume settings
- 2 do not disturb modes, including a Conversation Detection feature that will auto-snooze notifications while you’re listening to or engaging in discussions around you
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TITLE: In Zagier's one-sentence proof, why is S defined to be {(x,y,z)∈ℕ^3:x^2+4yz=p,p prime}?
QUESTION [4 upvotes]: I've looked at a very clear explanation of Zagier's proof (specifically, it can be found here:http://danielmath.wordpress.com/2012/12/26/one-sentence-proof/) but the first step still eludes me: why is S defined to be {(x,y,z)∈ℕ^3:x^2+4yz=p,p prime}? How does one arrive at that definition of S from the equation p=4k+1? I do not see where the three variables x,y,z came from, why they have to be a part of ℕ that is cubed,such that x^2+4yz=p - in essence, I really have no idea how this definition of S came to be.
Many, many thanks!
REPLY [0 votes]: Michael's response is as clear as you will get without getting too involved. But still, in "the clever bit", why not write $p=xy+4z^2$ or even $p=xy + 4 wz$? And in Zagier's original proof, although you don't mention it, the complicated piecewise-defined involution should be at least as mysterious as consideration of the set S.
More understanding can be gained through several different perspectives. Many are found in Elsholtz's survey here:
http://www.math.tugraz.at/~elsholtz/WWW/papers/papers30nathanson-new-address3.pdf
(He also gives a beautiful alternative one-sentence proof that uses geometry in the torus $\mathbb{Z}_p \times \mathbb{Z}_p$.) He shows, in particular, that if one searches for a linear involution to accomplish what Zagier's proof does, then Zagier's is the unique one with suitably "nice" properties.
Some of Elsholtz's survey is drawn from work of Dijkstra:
http://www.cs.utexas.edu/users/EWD/ewd11xx/EWD1154.PDF
On replacing the relation $p=x^2 + 4y^2$ by $p=x^2 + 4yz$, Dijkstra says: "we establish a one-to-one correspondence between the solutions of $p=x^2 + 4y^2$ and the fixed points of an involution. For the construction of that involutions, we do something with which every computer scientist is very familiar: replacing in a target relation -- here $p=x^2 + 4y^2$ -- something by a fresh variable”.
A perspective in terms of binary quadratic forms can be found here:
http://www.rzuser.uni-heidelberg.de/~hb3/publ/bf.pdf
where we see $p=x^2+4yz$ arise naturally when fixing the discriminant of a general binary quadratic form.
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TITLE: Irreducible representation intuition
QUESTION [0 upvotes]: Please explain this concept: Reducible and irreducible representations of a group, invariant subspaces of a group.
I suspect it will be easiest to understand with a finite group and a linear representation, but if there's extra insight to be gained by considering infinite, continuous, or non-linear groups, have at it.
I hope to understand things about the structure of the group itself, and I see linear algebra as merely a means to that end.
If possible, please give references to important related concepts.
In your answers, feel free to assume knowledge of the basics of group theory: Group axioms, Abelian/Nonabelian, invariant subgroups, homo/isomorphisms, related algebraic structures, etc.
Thanks
REPLY [2 votes]: A representation of a group is a homomorphism from the group $G$ to the group of transformations (the “automorphism group”) of some mathematical object $A$.
If $A$ is a vector space then its automorphism group is the group of linear transformations of $A$, and the representation is called a linear representation. If $A$ is a finite $n$ dimensional vector space over a field $K$ then the group representation is a homomorphism from $G$ to $GL(n,K)$, the group of $n$ dimensional invertible matrices over $K$.
If two representations $a$ and $b$ are related by an automorphism $c$ such that $a=cbc^{-1}$ then $a$ and $b$ are equivalent representations. For linear representations, this just means that we have chosen a different basis for the underlying vector space.
Representations can be combined to form more complex representations. For example, two $2$ dimensional linear representations can be combined to create a $4$ dimensional linear representation, with each component acting on a $2$ dimensional sub-space. Going in the opposite direction, some representations can be broken down into simpler representations. These are called reducible representations. Representations that cannot be broken down into simpler representations are called irreducible representations.
A set of inequivalent and irreducible representations of a group is an interesting object which can be used to investigate properties of the group itself.
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\begin{document}
\title{Catching homologies by geometric entropy}
\author{Domenico Felice}
\email{domenico.felice@unicam.it}
\affiliation{School of Science and Technology,
University of Camerino, I-62032 Camerino, Italy \\
INFN-Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italy}
\author{ Roberto Franzosi}
\email{roberto.franzosi@ino.it}
\affiliation{QSTAR and INO-CNR, largo Enrico Fermi 2, I-50125 Firenze, Italy}
\author{Stefano Mancini}
\email{stefano.mancini@unicam.it}
\affiliation{School of Science and Technology,
University of Camerino, I-62032 Camerino, Italy \\
INFN-Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italy}
\author{Marco Pettini}
\email{pettini@cpt.univ-mrs.fr}
\affiliation{ Aix-Marseille University, Marseille, France\\
CNRS Centre de Physique Th\'eorique UMR7332,
13288 Marseille, France}
\begin{abstract}
A geometric entropy is defined as the Riemannian volume of the parameter space of a statistical manifold associated with a given network. As such it can be a good candidate for measuring networks complexity. Here we investigate its ability to single out topological features of networks proceeding in a bottom-up manner: first we consider small size networks by analytical methods and then large size networks by numerical techniques. Two different classes of networks, the random graphs and the scale--free networks, are investigated computing their Betti numbers and then showing the capability of geometric entropy of detecting homologies.
\end{abstract}
\pacs{89.75.-k Complex systems; 02.40.-k Differential geometry and topology; 89.70.Cf Entropy}
\maketitle
\section{Introduction}
Common understanding identifies a \textit{network} as a set of items, called \textit{nodes} (or vertices), with connections between them, called \textit{links} (or edges) \cite{Newman}. Since many systems in the real word take the form of networks (also called \textit{graphs} in much of the mathematical literature), they are extensively studied in many branches of science, like, for instance, social, technological, biological and physical science \cite{Boccaletti06}. At the beginning, the study of networks was one of the fundamental topics in discrete mathematics: Euler is ascribed as the first providing a true proof in the theory of networks by its solution of the K\"onigsberg problem in 1735. Recently, also thanks to the availability of computers, the study of networks moved from the analysis of single small graphs and the properties of individual nodes and links within such graphs to the consideration of large-scale statistical properties of graphs. Thus, statistical methods became a prominent tool to quantify the degree of organization (\textit{complexity}) of large networks \cite{barabasi1}.
A typical approach in statistical mechanics of complex networks is the statistical ensemble. Such an approach is a natural extension of Erd\"os-R\'enyi ideas \cite{ER}. It has been performed through two basic ideas: the configuration space weight and the functional weight \cite{BBW06}. The first one is proportional to the uniform probability measure on the configuration space which accounts for the way to uniformly chose graphs in the configuration space. Whereas, functional weights depend on the network topologies and are chosen in order to address the statistical mechanics approach to networks different from the random graphs, which have some typical structures, like the \textit{small world} property \cite{storogatz}, the power-law degree distribution \cite{barabasi2}, the correlation of node degrees \cite{bollobas}, to name the most frequently addressed. During the last decade, several works have been inspired by this approach \cite{Bianconi}.
Recently, the techniques of statistical mechanics were complemented by new topological methods: a network is encoded through a simplicial complex which can be considered as a combinatorial version of a topological space whose properties can be studied from combinatorial, topological or algebraic points of view. Thus, regarding the mentioned topological aspects, different measures of simplicial complexes and of networks stemming from simplicial complexes can be defined. This provides a link between topological properties of simplicial complexes and statistical mechanics of networks from which simplicial complexes were constructed \cite{jstat}.
In this work we consider a geometric entropy which is inspired by microcanonical entropy of statistical mechanics and stems from Information Geometry (IG) \cite{FMP}. In particular, a Riemannian manifold (differentiable object) is associated to a network (discrete object) and the complexity measure is the logarithm of the volume of the manifold. More precisely, random variables are associated to each node of a network and their correlations are considered as weighted links among the nodes. The nature of these variables characterizes the network (for example, each node can contain energy, or information, or represent some internal parameter of a neuron in a neural network, or the concentration of a biomolecule in a complex network of biochemical reactions, and so on). The variables are assumed to be random either because of the difficulty of perfectly knowing their values or because of their intrinsic random dynamical properties. Thus, as it is customary for probabilistic graphs models \cite{keshav}, a joint probability mass function is associated to the description of the network. At this point, we assume Gaussian joint probability mass functions because of their tractability and since they are used extensively in many applications ranging from neural networks, to wireless communication, from proteins to electronic circuits, etc. Finally, the geometric complexity measure of networks is obtained by resorting to the afore mentioned relation between networks and joint probability mass functions and introducing in the space of these mass functions a Riemannian structure borrowed from information geometry \cite{amari}.
In addition to the statistical methods in network complexity we also consider topological methods by encoding a network into a \textit{clique graph} $C(G)$, that has the complete subgraphs as simplexes and the nodes of the graph (network) $G$ as its nodes so that it is essentially the complete subgraph complex. The maximal simplexes are given by the collection of nodes that make up the cliques of $G$. In particular, we are interested in the information about the topological space $C(G)$ stored in the number and type of holes it contains. So, we exploit algebraic topology tools in order to describe a network by means of its homology groups \cite{Carlsson}.
The ability of the geometric entropy to capture algebraic topological features of networks is investigated by a bottom-up approach. First, we consider networks with low number of nodes (\textit{small--size}) and given homology groups. Then, we compute the dimension of the homology groups of large--size networks. Hence, we compare the geometric entropy against the dimensions of homology groups (Betti numbers) revealing a clear detection of topological properties of the considered networks. In particular, when dealing with large--size networks, we consider random graphs and scale--free networks. According to the well--known transition in the appearance of a giant component\cite{ER,aiello2001}, a description of networks through their Betti numbers shows a clear correlation with the growth of the size of the largest components. This perfectly matches the behaviour of the geometric entropy \cite{Franzosi16} which in turn, when compared to Betti numbers of the the networks, clearly appears to probe relevant topological aspects of the networks.
The organization of the present paper is as follows. In Section \ref{sec2} we review some methods of the Algebraic Topology useful to describe topological properties of networks encoded in simplicial complexes. In Section \ref{sec3} we describe the geometric entropy stemming from both statistical methods and IG methods. In Section \ref{sec4} we compute the Homology groups of small--size networks as well as of large size networks within the ensembles of random graphs and scale--free networks. Then we compare the geometric entropy computed on these networks against their Betti numbers. Concluding remarks are given in Section \ref{sec5}.
\section{Basics of Algebraic Topology}\label{sec2}
In order to make the present work self contained, we start by reviewing some methods of combinatorial algebraic topology {by referring to} \cite{spanier}; these methods allow a topological characterization of networks. In particular, we focus on simplicial algebraic invariants; among them, we select homologies since they are easier to compute than, for example, {homotopy groups.}
\subsection{Simplicial Complexes}
Let $I=\{v_i\}_{i\in\NN}$ be a set of vertices (or nodes). A \textit{simplex} $s$ in $I$ with dimension $n$ is any its subset with cardinality equal to $n+1$, and it is called a $n$-simplex; in particular the \textit{empty set} is the only $-1$-simplex. A \textit{face} of a $n$-simplex $s:=\{v_0,v_1,\ldots,v_n\}$ is the simplex $s^\prime$ {whose} vertices consist of any nonempty subset of the $v_i$s; if $s^\prime$ is a $p$-simplex, with $p<n$, it is called a $p$-face of $s$. The subset needs not be a proper subset, so $s=\{v_0,v_1,\ldots,v_n\}$ is regarded as a face of itself.
A \textit{simplicial complex} $K$ consists of a set $I$ of vertices and a set $\{s\}$ of simplexes such that
(i) any set consisting of exactly one vertex is a simplex; (ii) any nonempty subset of a simplex is a simplex.
It follows from condition (i) that $0$-simplexes of $K$ correspond bijectively to vertices of $K$. {Analogously, from condition (ii) it follows} that any simplex is determined by its $0$-faces. Thus, we can identify $K$ {as the set of its simplexes, and a vertex of $K$ as} the $0$-simplex corresponding to it. For example, let $I=\ZZ/{n\ZZ}$ be the set of vertices, and consider the simplicial complex $P_n$ on $I$ with set of simplexes $\{i,i+1\},\,i\in I$. Intuitively, $P_n$ is a set of $n$ vertices and $n$ links (edges) among them. Therefore, $P_n$ is called the \textit{standard polygon} with $n$ edges. If, we add to it the $2$-simplexes $\{0,i,i+1\}$, for $i=1,\ldots,n-2$, we arrive at the simplicial complex $D_n$ called the \textit{standard polygonal disk}. Less formally, in order to obtain $D_n$ we add to the standard polygon $P_n$ the triangles that we can construct among triplets of the set of $n$ vertices. As an example, we draw the difference between a polygon and a polygonal disk in Figure \ref{polygon} when $n=5$.
\tikzstyle{every node}=[circle, draw, fill=green!50,
inner sep=0pt, minimum width=4pt]
\begin{figure}\centering
\vspace{0.2cm}
\begin{tabular}{c}
\begin{tikzpicture}[thick,scale=0.8]
\draw[fill=yellow!20] \foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} -- (\x+288:2) node{4} -- (\x+360:2) node{0}
(\x:2) -- (\x+144:2)
(\x:2) -- (\x+216:2)
};
\end{tikzpicture}
\end{tabular}\hspace{2.5cm}
\begin{tabular}{c}
\begin{tikzpicture}[thick,scale=0.8]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} -- (\x+288:2) node{4} -- (\x+360:2) node{0}
};
\end{tikzpicture}
\end{tabular}
\caption{(Left)A polygonal disk with $5$ nodes. (Right)A standard polygon with $5$ edges.} \label{polygon}
\end{figure}
The \textit{dimension} of a simplicial complex
$K$ {is $\mbox{dim} K=\sup \{\mbox{dim}\, s;s\in K\}$.
$K$ is said to be \mbox{finite} if it contains only a finite number of simplexes. In such a case}
$\mbox{dim} K<\infty$; however, if $\mbox{dim} K<\infty$, $K$ {needs not be finite.} Indeed, consider the simplicial complex $K$ with set of vertices $\ZZ$, and as the set of simplexes the edges $\{i,i+1\}$ as $i$ varies in $\ZZ$; then $K=\{\{i\};i\in\ZZ\}\cup\{\{i,i+1\};i\in\ZZ\}$. In this case, $K$ is not finite but $\mbox{dim} K=1$. A simplicial complex $K$, that we have abstractly defined upon a set of vertices $I=\{v_i\}_{i=1}^k$, can be also assigned by means of a {\it geometric realisation} by assuming that its vertices are points in $\RR^n$. For example, we get what is known as the natural realisation if we take $n = k + 1$ and $v_0 = e_1, v_1 = e_2,\ldots, v_k = e_{k+1}$, where the $e_i$ are the standard basis vectors in $\RR ^n$. We note that although $K$ may be $n$-dimensional, a realisation $K$ may not ``fit'' into $\RR^n$. Indeed, the standard polygon on $5$ vertices cannot be embedded into $\RR ^1$ even though its dimension amounts to $1$. For this reason, being networks abstract discrete objects, we rely on the combinatorial {approach avoiding any particular geometric realisation.}
{When a simplicial complex $K$ is finite, it is possible to introduce the \textit{Euler-Poincar\'e characteristic} $\chi(K)$ as the summation along the number of all its simplexes }
\begin{equation}
\label{E-P}
\chi(K)=1+(-\nu_{-1}+\nu_0-\nu_1+\nu_2-\ldots)=1+\sum_{p\in\ZZ}(-1)^p\,\nu_p,
\end{equation}
where $\nu_p$ is the number of $p$-simplexes. In particular, we have that $\chi(\emptyset)=0$.
\subsection{Paths and fundamental group}
Let $I$ be a set of vertices, a \textit{step} into a simplicial complex $K$ on $I$ is an element $q=(v,w)\in I\times I$ such that $\{v,w\}\in K$; $v$ is the \textit{initial point} and $w$ is the \textit{final point} of the step. Two steps $q_1,\,q_2$ are \textit{consecutive} if the final point of $q_1$ is the initial point of $q_2$. A \textit{path} is a sequence $\gamma=(q_1,\ldots,q_n)$ of consecutive steps. Two paths $\alpha=(\alpha_1,\ldots,\alpha_r)$ and $\beta=(\beta_1,\ldots,\beta_s)$ can be multiplied if $(\alpha_1,\ldots,\alpha_r,\beta_1,\ldots,\beta_s)$ is a path, i.e. $\alpha_r$ and $\beta_1$ are consecutive; such a path is called the product $\alpha*\beta$.
Consider $v,w,z\in I$ such that $\{v,w,z\}\in K$; we say that the path $((v,w),(w,z))$ is elementary contractible in the step $(v,z)$. By referring to the {Fig.\ref{polygon}-left the path $((2,1),(1,0))$ can be contracted to the path $(2,0)$, whereas there is no way to contract any paths in Fig.\ref{polygon}-right.} In general, if a path $\alpha$ has two consecutive steps $((v,w),(w,z))$, we can substitute them with the step $(v,z)$; in this way we obtain a new path $\alpha^\prime$, and we say that $\alpha^\prime$ is obtained from $\alpha$ by an \textit{elementary contraction}. Vice versa, we say that $\alpha$ is obtained from $\alpha^\prime$ by an \textit{elementary expansion}.
Two paths $\alpha$ and $\beta$ in $K$ are called \textit{homotopic} if it is possible to go from one to the other by means of a finite number of elementary expansions or contractions; in this case we write $\alpha\sim\beta$. The latter is an equivalence relation that preserves the initial and the final points; moreover the product ``$*$" passes to the quotient, meaning that if $\alpha\sim\alpha^\prime$ and $\beta\sim\beta^\prime$, then $\alpha*\beta\sim\alpha^\prime*\beta^\prime$. Consider now $i_0\in I$; a \textit{loop} with base point $i_0$ is a path in $K$ which starts from $i_0$ and ends at $i_0$. The set of the homotopy {classes of loops at $i_0$ forms a group $\pi_1(K,i_0)$ under multiplication $*$,} called the \textit{fundamental group} of $K$ at $i_0$.
\subsection{Chains, cycles and boundaries}
Given the usual set $I$ of vertices, an \textit{oriented} $p$-simplex of the simplicial complex $K$ on $I$ is a $(p+1)$-tuple $(v_0,v_1,\ldots,v_p)\in I^{p+1}$ such that $\{v_0,v_1,\ldots,v_p\}\in K$. For $p<0$ there are no oriented $p$-simplexes; for every vertex $v\in K$ there is a unique oriented $0$-simplex $(v)$. Then we define the $p$th module $S_p(K,\ZZ)$ of chains in $K$ as the free $\ZZ$-module generated by oriented $p$-simplexes. A $p$-chain is basically a linear combination of $p$-simplexes with coefficients in $\ZZ$. Then it straightforwardly follows that $S_p(K,\ZZ)=0$ for $p<0$.
Consider now the module--homomorphisms $\partial_p:S_p(K,\ZZ)\rightarrow S_{p-1}(K,\ZZ)$ for $p\geq 1$. Their values are uniquely determined on the $p$-simplexes. So, we can define
\begin{equation}
\label{hom}
\partial_p(v_0,v_1,\ldots,v_p)=\sum_{i=0}^p (-1)^i(v_0,\ldots,\hat{v_i},\ldots,v_p),
\end{equation}
where $(v_0,\ldots,\hat{v_i},\ldots,v_p)$ means the oriented $(p-1)$-simplex obtained by omitting $v_i$. {Actually,} the homomorphism $\partial_p$ is a boundary operator in the sense that it acts on a $p$--simplex by giving rise to its faces. It is not difficult to show that $\partial_p\partial_{p+1}=0$
{for all $p$.}
Let $w$ be an oriented $p$-chain in $K$, i.e. $w\in S_p(K,\ZZ)$; then, $w$ is called a $p$-\textit{cycle} if $\partial_p(w)=0$; $w$ is called a $p$-\textit{boundary} if there exists $z\in S_{p+1}(K)$ such that $w=\partial_{p+1}(z)$. The set of the $p$-cycles $Z_p(K,\ZZ)$, and the set of the $p$-boundaries $B_p(K,\ZZ)$ are submodules of $S_p(K,\ZZ)$, {and the relation
$B_p(K,\ZZ)\subset Z_p(K,\ZZ)$ holds true for $p\in\ZZ$.} Less formally, a cycle is a member of $B_p(K,\ZZ)$ if it ``bounds'' something contained in the simplicial complex $K$. For example, by referring to the polygonal disk $D_n$ of Fig. \ref{polygon}, we can see that the oriented chain $(1,0)+(0,2)+(2,1)$ is a boundary, while the chain $(1,0)+(0,4)+(4,3)$ is not.
\subsection{Homology groups}
Since $B_p(K,\ZZ)\subset Z_p(K,\ZZ)$ for all $p\in\ZZ$, we can consider the quotient module $H_p(K,\ZZ)=\ker \partial_p/\mbox{im}\,\partial_{p+1}=Z_p/B_p$ (called $p$th-module of homology). Intuitively, the construction of homology assumes that we are removing the cycles that are boundaries of higher dimension from the set of all $p$-cycles, so that the ones that remain carry information about $p$-dimensional holes of the simplicial complex.
If $H_p(K,\ZZ)$ is finitely generated (which is necessarily true if $K$ has finitely many simplexes) from the structure theorem {\cite{spanier}} it follows the $H_p(K,\ZZ)$ is isomorphic to the direct sum of a finite free $\ZZ$-module $H_p^\prime$ and a finite number of finite cyclic groups $\ZZ/n_1\ZZ\oplus\ZZ/n_2\ZZ\oplus\ldots\oplus\ZZ/n_\kappa\ZZ$, where $n_i$ divides $n_{i+1}$. Thus, the $\mbox{rank}(H_p(K,\ZZ))$ is defined as the number of basis elements of $H^\prime$ on $\ZZ$.
{Such a rank is also the $p${th} Betti number of $K$, i.e. $\beta_p:=\mbox{rank}(H_p(K,\ZZ))$,
hence the Euler-Poincar\'e characteristic {\eqref{E-P}} becomes} $\chi(H(K))=1+\sum_{p\in\ZZ}(-1)^p\,\beta_p$.
Let us now compute the lower homology groups. {Consider first an empty simplicial complex $K$}. In this case, it has only one $(-1)$-simplex, the empty set $\emptyset$; thus, the complex chain $S_{-1}(K,\ZZ)$ is isomorphic to $\ZZ$. In addition, $S_p(K,\ZZ)=(0)$ for $p\neq -1$. For these reasons $H_{-1}(\emptyset,\ZZ)\cong \ZZ$ and $H_p(K,\ZZ)=(0)$ for $p\neq -1$. {For non empty $K$ the homomorphism $\partial_0:S_0(K,\ZZ)\rightarrow S_{-1}(K,\ZZ)$ is surjective and $H_{-1}(K,\ZZ)=(0)$. Concerning $H_0(K,\ZZ)$, notice} that
$$\partial_0:S_0(K,\ZZ)\rightarrow S_{-1}(K,\ZZ)(\cong\ZZ)$$
is defined by
\begin{equation}
\label{boundaryoperator}
\partial_0(r_1\,i_1+r_2\,i_2+\ldots+r_N\,i_N)=r_1+\ldots+r_N,
\end{equation}
for any $i_1,i_2,\ldots,i_N\in\ZZ$. From \eqref{boundaryoperator} and the relation $\partial_0\partial_1\equiv 0$ it follows that, if $(i_0,i_1)$ is a $1$-simplex in $K$, then $\partial_0(i_0,i_1)=i_1-i_0$. For this reason $\partial_0(\sigma)=v_1-v_0$ when $\sigma$ is a path in $K$ from $v_0$ to $v_1$.
Let us now assume that $K$ is connected and $v_0\in K$. Then, every $0$-cycle $r_1\,i_1+r_2\,i_2+\ldots+r_N\,i_N$ can be written as $r_1\,(i_1-v_0)+r_2\,(i_2-v_0)+\ldots+r_N\,(i_N-v_0)$ and it follows that $H_0(K,\ZZ)=(0)$. {In contrast,} if $K$ is not connected and $V$ is the set of vertices one for any connected component, we have that $H_0(K,\ZZ)$ is the $\ZZ$-free module on the classes $v-v_0$ for $v\in V$. Hence, the notion of connectivity in $K$ is reflected on $H_0(K,\ZZ)$, the dimension of which, that is the Betti number $\beta_0$, counts the number of connected components of a simplicial complex $K$.
Again, consider a connected simplicial complex $K$ and ad vertex $v_0\in K$. An homomorphism from the fundamental group of homotopy to the first homology group
$$\nu:\pi_1(K,\ZZ)\rightarrow H_1(K,\ZZ)$$
can be obtained {via} the morphism
$$\sigma:P_n\rightarrow K,$$
where $P_n$ is the $n$-edges closed polygon. In this way, the homomorphism $\nu$ is defined as follows
$$\nu\left([P_n]_{\pi_1}\right)= \sigma_*[P_n]_{\pi_1}:=[\sigma(P_n)]_{H_1},$$
where $[P_n]_{\pi_1}$ is the class of $P_n$ in $\pi_1(K,\ZZ)$ and $[\sigma(P_n)]_{H_1}$ is the homology class of $\sigma(P_n)$. It is not difficult to prove that $\nu$ is surjective and its kernel is given by the commutators $[\alpha,\beta]=\alpha*\beta*\alpha^{-1}*\beta^{-1}$, $\alpha,\beta\in\pi_1(K,\ZZ)$. Hence, the homology module $H_1(K,\ZZ)$ represents the classes of loops in the simplicial complex $K$. The same methods can be applied to any connected components of $K$ whenever it is not connected. Therefore, the Betti number $\beta_1$ counts the number of loops
{(one dimensional holes)} that are present in a simplicial complex $K$.
Finally, as far as the module $H_2(K,\ZZ)$ is concerned, consider a $1$-connected simplicial complex $K$, i.e. it is connected and simply connected. Then it is possible to represent any classes in $H_2(K,\ZZ)$ by means of the geometric representation $\phi:P\rightarrow K$, where $P$ is a $2$-sphere. Following the idea carried out for the $H_1(K,\ZZ)$, we can say that the homology module $H_2(K,\ZZ)$ characterises the voids inside the simplicial complex $K$.
\section{Statistical models and geometric networks complexity measure}\label{sec3}
We now start describing the geometric entropy that was firstly introduced in \cite{FMP} and further investigated in \cite{Franzosi15}.
\\
\
Consider $n$ \textit{real} random variables (r.v.) $X_1,\ldots,X_n$ with joint probability distribution $p(x;\theta)$ given by the $n$-variate Gaussian density function
\begin{equation}
\label{Gaussian}
p(x;\theta)=\frac{\exp\Big[-\frac{1}{2}\,x^t\,C^{-1}(\theta)\,x\Big]}{\sqrt{(2\pi)^n\,\det\, C(\theta)}},
\end{equation}
which is characterized by $m$ \textit{real} parameters $\theta^1,\ldots,\theta^m$, i.e. the entries of the covariance matrix $C(\theta)$. Here, $t$ is the transposition and $x=(x_1,\ldots,x_n)\in\RR^n$ is the vector of values that $X_1,\ldots,X_N$ take on a sampling space $\Omega$. In addition, we assume mean-values being zero.
Consider now the collection of $n$-variate Gaussian density functions
\begin{equation}
\label{statmodel}
{\cal P}=\{p_\theta= p(x;\theta);\theta\in\Theta\subset\RR^m\},
\end{equation}
where $p(x;\theta)$ is as in Eq. \eqref{Gaussian} and $\Theta:=\{\theta\in\RR^m;C(\theta)>0\}$. So defined, $\cal P$ is an $m$-dimensional statistical model on $\RR^n$. Since a parameter $\theta\in\Theta$ uniquely describes distribution $p_\theta$, the mapping $\varphi:{\cal P}\rightarrow \RR^m$ defined by $\varphi(p_\theta)=\theta$ is one-to-one. We can thereby consider $\varphi=[\theta^i]$ as a system of local coordinates for $\cal P$. Hence, assuming parametrizations which are $C^\infty$ we can turn $\cal P$ into a differentiable manifold, that is called a \textit{statistical manifold} \cite{amari}.
Given $\theta\in\Theta$, the Fisher information matrix of $\cal P$ at $\theta$ is the $m\times m$ matrix $G(\theta)=[g_{ij}]$, where the $ij$ entry is defined by
\begin{equation}
\label{gFR}
g_{ij}(\theta):=\int_{\RR^n}dx\,p(x;\theta)\,\partial_i\log p(x;\theta)\partial_j\log p(x;\theta),
\end{equation}
with $\partial_i$ standing for $\frac{\partial}{\partial \theta^i}$. The matrix $G(\theta)$ is symmetric and positive semidefinite \cite{amari}. From here on, we assume it is positive definite; in such a way, $\Theta$ can be endowed with the proper Riemannian metric $g(\theta)=\sum_{i,j=1}^m g_{ij}\ d\theta^i\otimes d\theta^j$ and the manifold ${\cal M}:=(\Theta,g(\theta))$ is a Riemannian manifold.
From Eq. \eqref{Gaussian} the integral in \eqref{gFR} turns out to be a Gaussian one which is easily tractable. If, in addition, we assume that non-diagonal entries $c_{ij}$ of the covariance matrix $C(\theta)$ can take only $0$ or $1$ values, then an explicit analytical relation holds between entries of the matrix $G(\theta)$ and those of the matrix $C(\theta)$ \cite{FMP}, and is given by
\begin{equation}
\label{analytic}
g_{ij}=\frac{1}{2}(c_{ij}^{-1})^2,
\end{equation}
where $c_{ij}^{-1}$ is the $ij$ entry of the inverse of the covariance matrix $C(\theta)$.
As usual in mathematics, a geometric object is endowed with an over structure in order to employ more efficient tools to describe it (e.g. bundles over manifolds, coverings over topological spaces, and so on). Likewise, we want to endow a network (a discrete system) with a Riemannian manifold (a differentiable and continuous system). Among all the probabilistic methods the random walk one \cite{Noh04} allows a geometric approach through the Green function giving rise to a metric \cite{Mathieu08}. However, our geometric approach is different since, beside the adjacency matrix also the variances of a Gaussian distribution of random variables are taken into account. In fact, we basically exploit two elements: one is the functional relation in Eq. \eqref{analytic} between Fisher matrix $G(\theta)$ and Covariance matrix $C(\theta)$; the other one is the \textit{adjacency matrix} $A$ of a network.
First of all, we interpret random variables $X_1,\ldots,X_n$ as sitting on vertices of a network which is assumed a simple and undirected graph. The \textit{bare} system is assumed as a network without connections among the vertices. In this case we can consider $X_1,\ldots,X_n$ statistically independent. So, their covariance matrix is the $n\times n$ diagonal matrix $C_0(\theta)$ with entries given by
$$
(c_0)_{ii}:=\theta^i=\int_{\RR^n}\,dx\,p(x;\theta)x_i^2,\; i=1,\ldots,n,
$$
where $p(x;\theta)$ is given by \eqref{Gaussian}.
The parameter space reads as $\Theta_0=\{\theta^i>0;\,i=1,\ldots,n\}$. Furthermore, from Eq. \eqref{analytic} it follows that the Fisher information matrix is the diagonal $n\times n$ matrix $G_0=\frac{1}{2}\left[\left(\frac{1}{\theta^i}\right)^2\right]_{ii}$, for $i=1,\ldots,n$ and $\theta^i\in\Theta_0$. Finally, we can associate to the bare network the statistical Riemannian manifold ${\cal M}=(\Theta_0,g_0)$, with
\begin{equation}
\label{g0}
\Theta_0=\{\theta^i>0;\,i=1,\ldots,n\},\quad g_0=\frac{1}{2}\sum_{i=1}^n \Big(\frac{1}{\theta^i}\Big)^2\,d\theta^i\otimes d\theta^i.
\end{equation}
At this point, in order to take into account the possible connections among vertices of a network, we consider its adjacency matrix $A$. Then, let the map $\psi_{\theta}:A(n,\RR)\rightarrow GL(n,\RR)$ be defined by
\begin{equation}
\label{psi}
\psi_{\theta}(A):=C_0(\theta)+A,
\end{equation}
where $A(n,\RR)$ denotes the set of symmetric $n\times n$ matrices over $\RR$ with vanishing diagonal elements that can represent any simple undirected graph. The manifold associated to the network is the Riemannian manifold $\widetilde{M}=(\widetilde{\Theta},\widetilde{g})$ given by
\begin{equation}
\label{paramvary}
\widetilde{\Theta}:=\{\theta\in\Theta_0;\,\psi_{\theta}(A) \,\mbox{is positive definite}\}
\end{equation}
and $\widetilde{g}=\sum_{ij}\,\widetilde{g}_{ij}\,d\theta^i\otimes d\theta^j$ with components
\begin{equation}
\label{gvary}
\widetilde{g}_{ij}=\frac{1}{2}\Big(\psi_{\theta}(A)_{ij}^{-1}\Big)^2
\end{equation}
where $\psi_{\theta}(A)_{ij}^{-1}$ is $ij$ entry of inverse of the invertible matrix $\psi_{\theta}(A)$.
Hence, given a network with adjacency matrix $A$ and associated manifold $\widetilde{M}=(\widetilde{\Theta},\widetilde{g})$, we put forward the following network complexity measure \cite{Franzosi15},
\begin{equation}
\label{entropy}
\mathcal{S}=\ln{\cal V}(A),
\end{equation}
where ${\cal V}(A)$ is the volume of the manifold $\widetilde{M}$ obtained from the volume element
\begin{equation}
\label{volumeform}
\nu_{\widetilde{g}}=\sqrt{\det\widetilde{g}(\theta)}\,d\theta^1\,\wedge\ldots\wedge\,d\theta^n.
\end{equation}
Unfortunately, in such a way ${\cal V}(A)$ is ill-defined since the parameter space $\widetilde{\Theta}$ is not compact and since $\det\widetilde{g}(\theta)$ diverges as $\det\psi_{\theta}(A)$ approaches to zero for some $\theta^i$. Thus, as it is usual \cite{Leibb75}, we regularize it as follows
\begin{equation}
\label{regular}
{\cal V}(A)=\int_{\widetilde{\Theta}}{\cal R}(\psi_{\theta}(A))\,\nu_{\widetilde{g}},
\end{equation}
where ${\cal R}(\psi_{\theta}(A))$ is any suitable regularizing function. This should be a kind of compactification of the parameter space and with the excision of those sets of $\theta^i$ values which make $\det\widetilde{g}(\theta)$ divergent.
\section{Homology groups and geometric--entropy complexity of networks}\label{sec4}
In this section we apply the methods so far described, i.e. we compute the homology groups of given networks and compare them against the geometric--entropy values of the same networks. Actually, we refer to the dimensions of those groups via the Betti numbers and work out a relation between them and the geometric--entropy. In such a way, we show a clear detection of this kind of network topological features by means of the entropy ${\cal S}$ of the Eq. \eqref{entropy}.
The aim of this work consists of checking to what extent the geometric--entropy ${\cal S}$ can be sensitive to some topological property of networks beyond its effectiveness beforehand checked as a network complexity measure \cite{Franzosi16}.
In order to get well-grounded insights about the relation between the Betti numbers and the values of ${\cal S}$, we firstly consider low--size graphs upon which analytical methods can be worked out. Once that is set up, we investigate large--size networks particularly focusing on two graph ensembles, namely the random graphs and the scale--free networks.
\subsection{Analytical Results}
Let us consider five nodes graphs for our preliminary investigation. The clique graph interpretation thereby ascribes the simplicial complex $K_4$ as a $4$--simplex, i.e. a $5$ nodes fully connected network. In order to show how the entropy ${\cal S}$ works, we refer to two chains within $K_4$ as in Tab. \ref{Tab1}.
\begin{table}
[ht] \caption{The value of $\mathcal{S}$ for networks with five nodes against topological dimension}\label{Tab1}
\vspace{0.2cm}
\begin{tabular}{|cll|cl|cl|}
\hline Network & & & & \mbox{dim} & ${\cal S}$&\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} (\x+144:2) node{2} (\x+216:2) node{3} (\x+288:2) node{4} (\x+360:2) node{0}
};
\end{tikzpicture} & & & & 1& $0.4012$&\\
\hline
\end{tabular}
\begin{tabular}{|cll|cl|cl|}
\hline Network & & & & \mbox{dim} & ${\cal S}$ &\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw[fill=yellow!20]\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} (\x+216:2) node{3} (\x+288:2) node{4} (\x+360:2) node{0}
(\x+144:2) node{2} -- (\x:2) node{0}};
\end{tikzpicture} & & & & 2& $0.4179$&\\
\hline
\end{tabular}
\end{table}
The network on the left side of Tab. \ref{Tab1} is a chain with one $1$-simplex and three $0$-simplexes; from relation \eqref{dimension} it follows that its topological dimension is $1$. Whereas, the network on the right side of Tab. \ref{Tab1} is a chain with one $2$-simplex and two $0$-simplexes and from \eqref{dimension} we have that its topological dimension is $2$.
According to graph theory \cite{godsil}, the adjacency matrices of these networks are
\begin{equation}
\label{adj1-2}
A_1=\left(\begin{array}{ccccc}
0&1&0&0&0\\
1&0&0&0&0\\
0&0&0&0&0\\
0&0&0&0&0\\
0&0&0&0&0
\end{array}\right),
\quad
A_2=\left(\begin{array}{ccccc}
0&1&1&0&0\\
1&0&1&0&0\\
1&1&0&0&0\\
0&0&0&0&0\\
0&0&0&0&0
\end{array}\right),
\end{equation}
respectively. Consider now $5$ independent random variables $X_1,\ldots,X_5$ sitting on the five completely disconnected nodes of the network, with Gaussian joint probability distribution given by
\begin{equation}
\label{PxT5}
p(x;\theta)=\frac{\exp\left[-\frac{1}{2} x^t C^{-1}_0(\theta) x\right]}{\sqrt{(2\pi)^5\det C_0(\theta)}},
\end{equation}
where $x\in\RR ^5$ and covariance matrix $C_0(\theta)$ is given by
\begin{equation}
\label{cov0}
C_0=\left(\begin{array}{ccccc}
\theta^1&0&0&0&0\\
0&\theta^2&0&0&0\\
0&0&\theta^3&0&0\\
0&0&0&\theta^4&0\\
0&0&0&0&\theta^5
\end{array}\right),
\quad \theta^i=\int_{\RR^5 } dx\ p(x;\theta)\ x_i^2,\quad \forall i=1,\ldots,5.
\end{equation}
From Eq. \eqref{psi} we obtain that
\begin{equation}
\label{psi1-2}
\psi_{\theta}(A_1)=\left(\begin{array}{ccccc}
\theta^1&1&0&0&0\\
1&\theta^2&0&0&0\\
0&0&\theta^3&0&0\\
0&0&0&\theta^4&0\\
0&0&0&0&\theta^5
\end{array}\right),
\quad
\psi_{\theta}(A_2)=\left(\begin{array}{ccccc}
\theta^1&1&1&0&0\\
1&\theta^2&1&0&0\\
1&1&\theta^3&0&0\\
0&0&0&\theta^4&0\\
0&0&0&0&\theta^5
\end{array}\right).
\end{equation}
In order to associate a parameter space to each network, the matrices $\psi_{\theta}(A_1)$ and $\psi_{\theta}(A_2)$ must be positive definite, as indicated by Eq. \eqref{paramvary}. Such a condition is fulfilled by imposing that each main minor of the matrix has positive determinant. Thereby, we arrive at
\begin{eqnarray}
\label{parvary1-2}
&&\widetilde{\Theta}_1=\left\{\theta=(\theta^1,\ldots,\theta^5); \theta^1>0,\theta^1\theta^2>1,\theta^3>0,\theta^4>0,\theta^5>0\right\}\nonumber\\
\nonumber\\
&&\widetilde{\Theta}_2=\left\{\theta=(\theta^1,\ldots,\theta^5); \theta^1>0,\theta^1\theta^2>1,\theta^3>\frac{\theta^1+\theta^2-2}{\theta^1\theta^2-1},\theta^4>0,\theta^5>0\right\}.
\end{eqnarray}
Moreover, from Eq. \eqref{gvary} we endow $\widetilde{\Theta}_1$ and $\widetilde{\Theta}_2$, defined in \eqref{parvary1-2}, with Riemannian metrics, hence obtaining ${\cal M}_1=(\widetilde{\Theta}_1,\widetilde{g}_1) $ and ${\cal M}_2=(\widetilde{\Theta}_2,\widetilde{g}_2) $ as manifolds associated to the networks. Here, $\widetilde{g}_i$ and $\widetilde{g}_2$ are given by
\begin{eqnarray}
\label{gvary1-2}
&&\widetilde{g}_1=\frac{\left(\theta^2\right)^2d\theta^1\otimes d\theta^1+\left(\theta^1\right)^2d\theta^2\otimes d\theta^2}{2\left(\theta^1\theta^2-1\right)^2}+\frac{d\theta^4\otimes d\theta^4}{2\left(\theta^4\right)^2}+\frac{d\theta^5\otimes d\theta^5}{2\left(\theta^5\right)^2}+\frac{d\theta^1\otimes d\theta^2}{\left(\theta^1\theta^2-1\right)^2}\nonumber\\
\nonumber\\
&&\widetilde{g}_2=\frac{\left(\theta^2\theta^3-1\right)^2d\theta^1\otimes d\theta^1+\left(\theta^1\theta^3-1\right)^2 d\theta^2\otimes d\theta^2+\left(\theta^1\theta^2-1\right)^2 d\theta^3\otimes d\theta^3}{2\left(\theta^1(\theta^2\theta^3-1\right)-\theta^2-\theta^3+2)^2}\nonumber\\
&&+\frac{d\theta^4\otimes d\theta^4}{2\left(\theta^4\right)^2}+\frac{d\theta^5\otimes d\theta^5}{2\left(\theta^5\right)^2}+\frac{(1-\theta^3)^2d\theta^1\otimes d\theta^2+(1-\theta^2)^2d\theta^1\otimes d\theta^3}{\left(\theta^1(\theta^2\theta^3-1\right)-\theta^2-\theta^3+2)^2}.
\end{eqnarray}
The next ingredients necessary to compute the geometric--entropy values against the two networks are the volume elements computed through Eq. \eqref{volumeform}. However, as we can see from the analytical expressions for $\det\widetilde{g}_1$ and $\det\widetilde{g}_2$,
\begin{eqnarray}\label{det1-2}
&&\det\widetilde{g}_1=\frac{1+\theta^1\theta^2}{32\left(\theta^3\theta^4\theta^5\right)^2\left(\theta^1\theta^2-1\right)^3}\nonumber\\
\nonumber\\
&&\det\widetilde{g}_2=\frac{\left(\theta^1\right)^2\left(\left(\theta^2\theta^3\right)^2-1\right)+2\theta^1\left(\theta^2+\theta^3-2\theta^2\theta^3\right)^2-\left(\theta^2-\theta^3\right)^2}{32\left(\theta^4\theta^5\left(\theta^1+\theta^2+\theta^3-\theta^1\theta^2\theta^3-2\right)^2\right)^2},
\end{eqnarray}
it turns out that both ${\cal V}(A_1)$ and ${\cal V}(A_2)$ are ill-defined. In fact, on the one side, the numerator of $\det\widetilde{g}_i(i=1,2)$ diverges as $\theta^j$ goes to infinity for some $j\in\{1,\ldots,5\}$; on the other side, the denominator of $\det\widetilde{g}_i$ goes to zero as $\theta^j$ approaches the lower bound of $\widetilde{\Theta}_i(i=1,2)$ for some $j\in\{1,\ldots,5\}$.
In order to overcome this difficulty, we introduce a regularizing function ${\cal R}(\psi_{\theta}(A_{i}))$. This issue has been widely tackled in the literature (see for instance Ref. \cite{Leibb75}). As far as our approach is concerned, the aim is to make meaningful the following integrals
\begin{equation}\label{entropy1-2}
\int_{\widetilde{\Theta}_1}\ {\cal R}(\psi_{\theta}(A_{1}))\ \nu_{\widetilde{g}_1},\quad \int_{\widetilde{\Theta}_2}\ {\cal R}(\psi_{\theta}(A_{2}))\ \nu_{\widetilde{g}_2},
\end{equation}
where $\nu_{\widetilde{g}_1}=\sqrt{\det\widetilde{g}_1}\ d\theta^1\wedge\ldots\wedge d\theta^5$ and $\nu_{\widetilde{g}_2}=\sqrt{\det\widetilde{g}_2}\ d\theta^1\wedge\ldots\wedge d\theta^5$. Supported by a general result (see Corollary 2 in \cite{Felice17}) we consider the following regularizing function,
\begin{equation}\label{regf}
{\cal R}(\psi_{\theta}(A_{i})):=\left(H(h-\mbox{Tr}(C_0))+H(\mbox{Tr}(C_0)-h)\ e^{-\mbox{Tr}(C_0)}\right)\log\left(1+\left(\det\psi_{\theta}(A_{i})\right)^5\right).
\end{equation}
Here, $H(\cdot)$ denotes the Heaviside step function and $\mbox{Tr}$ is the trace operator, while $h$ is a \textit{real} number accounting for a finite range of $\theta^j$s values where integrals in \eqref{entropy1-2} are substantial. Regularizing function in \eqref{regf} depends on structure of networks through adjacency matrix $A_i$ and cures all the possible divergences of integrals \eqref{entropy1-2}, due to the functional forms in \eqref{det1-2}. Indeed, if $\theta^j$ goes to infinity for some $j=1,\ldots,5$, then the integrand is killed to zero by function $\left(H(h-\mbox{Tr}(C_0))+H(\mbox{Tr}(C_0)-h)\ e^{-\mbox{Tr}(C_0)}\right)$. Moreover, if $\det\psi_{\theta}(A_{i})$ goes to zero then the singularity is removed by the $\log\left(1+\left(\det\psi_{\theta}(A_{i})\right)^5\right)$ exploiting the well-known relation $\lim_{x\rightarrow 0}\frac{\log(1+x)}{x}=1$.
We are now ready to compute the values of the integrals in \eqref{entropy1-2} through the regularizing functions as in \eqref{regf} which are shown in Tab. \ref{Tab1}. The difference of entropic values between the $1$-dimensional chain and the $2$-dimensional chain is a particular result of a more general one \cite{FMP}. In addition, it is worth remarking that integrals in \eqref{entropy1-2} do not depend on the particular label permutation of the network \cite{FMP}. Hence, if two networks are isomorphic, then the values of ${\cal S}$ as in \eqref{entropy} coincide.
Actually, two isomorphic networks, intended as clique graphs, have the same topological features; thereby, they have the same homotopy and homology groups. As a consequence, whenever the Betti numbers $\beta_0$ and $\beta_1$ are different, then the networks are not isomorphic. In Tab. \ref{Tab2} we show the geometric--entropic values of ${\cal S}$ as in \eqref{entropy} computed for several $5$--nodes networks. In particular, we point out that ${\cal S}$ reflects the differences of their first homology groups. Indeed, by following the values on $\beta_0$ in Tab. \ref{Tab2}, we can see that the larger the number of connected components of a network, the lower the value of the entropy \eqref{entropy}. On the contrary, according to the values of $\beta_1$ in Tab. \ref{Tab2}, the larger the number of cycles the more complex the network. In addition, cycles mostly influence the complexity with respect to topological dimension as it follows by comparing the values of the entropy \eqref{entropy} of the last two networks on the right side of Tab. \ref{Tab2}.
\begin{table}
[ht] \caption{The value of $\mathcal{S}$ for networks with five nodes against topological dimension and Betti numbers $\beta_0$ and $\beta_1$} \label{Tab2}
\vspace{0.2cm}
\begin{tabular}{|clll|c|c|c|c|}
\hline Network & & & & \mbox{dim}& $\beta_0$ & $\beta_1$ & $\mathcal{S}$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} (\x+144:2) node{2} (\x+216:2) node{3} (\x+288:2) node{4} (\x+360:2) node{0}
};
\end{tikzpicture} & & & & $1$& $4$ & $0$ & $0.4012$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} (\x+144:2) node{2} (\x+216:2) node{3} (\x+288:2) node{4} -- (\x+360:2) node{0}
};
\end{tikzpicture} & & & & $1$& $3$ & $0$ & $1.5258$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} (\x+144:2) node{2} -- (\x+216:2) node{3} (\x+288:2) node{4} -- (\x+360:2) node{0}
};
\end{tikzpicture} & & & & $1$& $2$ & $0$ & $1.7153$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} (\x+288:2) node{4} (\x+360:2) node{0}
};
\end{tikzpicture} & & & & $1$& $2$ & $0$ &$2.0849$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} -- (\x+288:2) node{4} (\x+360:2) node{0}
};
\end{tikzpicture} & & & &$1$& $1$ & $0$ & $2.8739$\\
\hline
\end{tabular}
\begin{tabular}{|clll|c|c|c|c|}
\hline Network & & & & \mbox{dim}& $\beta_0$ & $\beta_1$ & $\mathcal{S}$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} (\x+288:2) node{4} (\x+360:2) node{0}
(\x:2) node{0} -- (\x+216:2) node{3}};
\end{tikzpicture} & & & &$1$& $2$ & $1$ & $3.0419$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} -- (\x+288:2) node{4} -- (\x+360:2) node{0}
};
\end{tikzpicture} & & & &$1$& $1$ & $1$ & $3.2699$\\
\hline
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw[fill=yellow!20] \foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} (\x+288:2) node{3} -- (\x+216:2) node{4} (\x+360:2) node{0}
(\x+144:2) node{2} -- (\x:2) node{0}
(\x+72:2) node{1} (\x+288:2) node{3}
(\x+144:2) node{2} -- (\x+216:2) node{4}
(\x+288:2) node{3} -- (\x:2) node{0} };
\end{tikzpicture} & & & &$2$& $1$ & $1$ & $4.7617$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw[fill=yellow!20] \foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+252:1) node{3} (\x+252:2) node{4} -- (\x+360:2) node{0}
(\x+72:2) node{1} -- (\x+252:1) node{3}
(\x+144:2) node{2} -- (\x+252:2) node{4}
(\x+252:1) node{3} -- (\x:2) node{0} };
\end{tikzpicture} & & & &$2$& $1$ & $1$ & $5.4758$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+45:0) node{3} (\x+252:2) node{4} -- (\x+360:2) node{0}
(\x+144:2) node{2} -- (\x+252:2) node{4}
(\x+45:0) node{3} -- (\x:2) node{0} };
\end{tikzpicture} & & & &$1$& $1$ & $2$ & $6.0841$\\
\hline
\end{tabular}
\end{table}
\bigskip
In Tab. \ref{Tab3} we can see that networks with the same Betti numbers $\beta_0$ and $\beta_1$ do not have the same values of ${\cal S}$. This is in agreement with results in \cite{Franzosi16} where it has been shown that networks with different degree distributions do not have the same complexity computed by ${\cal S}$. Hence, characterizing networks complexity through their first homology groups is not enough. Beyond these topological features, it is necessary to consider also other aspects as it is well--known in the vast literature around. For this reason, the fact that ${\cal S}$ takes different values for networks having the same Betti numbers $\beta_0$ and $\beta_1$ is not a validity restriction of the geometric entropy.
\begin{table}
[ht] \caption{The value of $\mathcal{S}$ for networks with five nodes and same homology groups}\label{Tab3}
\vspace{0.2cm}
\begin{tabular}{|clll|c|c|c|c|}
\hline Network & & & & \mbox{dim}& $\beta_0$ & $\beta_1$ & $\mathcal{S}$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} (\x+144:2) node{2} -- (\x+216:2) node{3} (\x+288:2) node{4} -- (\x+360:2) node{0}
};
\end{tikzpicture} & & & & $1$& $2$ & $0$ & $1.7153$\\
\hline
\end{tabular}
\begin{tabular}{|clll|c|c|c|c|}
\hline Network & & & & \mbox{dim}& $\beta_0$ & $\beta_1$ & $\mathcal{S}$\\
\hline
\begin{tikzpicture}[thick,scale=0.5]
\draw\foreach \x in {0}
{
(\x:2) node{0} -- (\x+72:2) node{1} -- (\x+144:2) node{2} -- (\x+216:2) node{3} (\x+288:2) node{4} (\x+360:2) node{0}
};
\end{tikzpicture} & & & & $1$& $2$ & $0$ &$2.0849$\\
\hline
\end{tabular}
\end{table}
\subsection{Numerical Results}
We now investigate the behaviour of the geometric--entropy ${\cal S}$, given in \eqref{entropy}, by comparing it with respect to the behaviour of the Betti numbers computed for two ensembles of networks \cite{Bianconi}, namely the random graphs and the scale--free networks.
The numerical computation of ${\cal S}$ has been performed on $200$--nodes networks by exploiting the functional form \eqref{gvary} of the components of the varied Fisher--Rao metric $\widetilde{g}$. Then, the volume regularization has been obtained at first by restricting the manifold support ${\widetilde{\Theta}}\subset \mathbb{R}^n$ to an hypercube, and we worked out Monte Carlo estimates of the average
$
\left\langle\sqrt{\det \widetilde{g}} \right\rangle =\int \sqrt{\det \widetilde{g}}\;d\theta^1\wedge\ldots\wedge d\theta^n/\int \;d\theta^1\wedge\ldots\wedge d\theta^n
$
by generating Markov chains inside ${\widetilde{\Theta}}$. In this case, the number of random configurations considered varies between $10^4$ and $10^6$. Finally, the regularization procedure of the volume is obtained by excluding those points where the value of $\sqrt{\det \widetilde{g}}$ exceeds $10^{308}$ (the numerical overflow limit of the computers used) \cite{Franzosi15}.
One of the basic models of random graphs is the uniform random graph $\mathbb{G}(n,k)$. This is devised by choosing with uniform probability a graph from the set of all the graphs having $n$ vertices and $k$ edges, with $k$ a nonnegative integer \cite{Lucz}. Scale--free networks can be obtained as special cases of random graphs with a given degree distribution showing thereof a power--law degree distribution. These are described by two parameters, $\alpha$ and $\gamma$ , which define the size and the density of a network; hence, given the number of nodes $n$ with degree $d$, these models, denoted $\mathbb{G}_{\alpha,\gamma}$, assigns a uniform probability to all graphs with $n=e^{\alpha} d^{-\gamma}$ \cite{Boccaletti06}.
In Fig. \ref{RG} we report the behaviour of ${\cal S}/n$ of $\mathbb{G}(n,k)$ vs $k/n$ for a fixed value of $n$, namely $n=200$, together with the behaviour of the Betti numbers $\beta_0$ and $\beta_1$ of $\mathbb{G}(n,k)$ \cite{Vittorio}. By interpreting $\mathbb{G}(n,k)$ as clique graphs, we can see a perfect correlation between ${\cal S}/n$ and the Betti number $\beta_0$. This is not surprising as $\beta_0$ reflects the number of connected components of $\mathbb{G}(n,k)$ and the appearance of a giant component is accounted for $\mathbb{G}(n,k)$ when $k/n>0.5$. Whereas the correlation of ${\cal S}$ with the Betti number $\beta_1$ of $\mathbb{G}(n,k)$ is subtler to interpret. Indeed, $\beta_1$ rapidly increases its value as $k/n$ increases, contrary to ${\cal S}/n$ which shows a saturation when the numbers of $k$ is large. Actually, $\beta_1$ counts the cycles of $\mathbb{G}(n,k)$ and if on one side $\beta_1$ equals zero when $k\leq n/2$ reflecting a well--known theoretical result \cite{Lucz}, on the other side it is not enough for completely describing the topology of $\mathbb{G}(n,k)$ when $k$ is larger than $5/2\ n $. However, the geometric--entropy ${\cal S}$ properly correlates with the Betti numbers $\beta_0$ and $\beta_1$ when $k\leq 1.5\ n$ and the only homology groups of $\mathbb{G}(n,k)$ are $H_0$ and $H_1$.
\begin{figure}[ht]\centering
\includegraphics[scale=.75]{RG}
\caption[10 pt]{Behaviours of the geometric--entropy ${\cal S}/n$ and the Betti numbers $\beta_0$, $\beta_1$ of $\mathbb{G}(n,k)$ as functions of the number $k$ of randomly chosen links of weights equal to $1$.}\label{RG}
\end{figure}
The scale--free network ensemble is recovered in the case of networks with a finite number of cycles \cite{Bianconi07}. In order to highlight the connection of the geometric--entropy ${\cal S}$ of Eq. \eqref{entropy} with the topology of the power--law random graph model $\mathbb{G}_{\alpha,\gamma}$ intended as a clique graph we have proceeded as follows. We considered networks of $n = 200$ nodes for which, without loss of generality, we set $\alpha=0$. For each value of $\gamma$ , we selected $10$ different realisations of the networks,
each realisation having the same value of $k/n$. Actually, because of the practical difficulty of getting different
realisations of a scale--free network with exactly the same value of $k/n$ at different $\gamma$ values, we accepted a spread of values in the range $0.7-0.85$.
In Fig. \ref{SF} we report the behaviour of the geometric--entropy ${\cal S}/n$ of the power--law random graphs $\mathbb{G}_{\alpha,\gamma}(n,k)$ when $\alpha=0$, $n=200$, and the exponent $\gamma$ is in the range $2.3 < \gamma < 4.5$ together with the behaviours of the Betti numbers of the same networks $\mathbb{G}_{\alpha,\gamma}(n,k)$ \cite{Vittorio}. The pattern of ${\cal S}/n$ displays a clear correlation with the Betti number $\beta_1$. In addition, in the range of $\gamma$ considered here, the only topological features of $\mathbb{G}_{\alpha,\gamma}(n,k)$ consist of the homology groups $H_0$ and $H_1$. Hence, a tight correlation between the geometric--entropy ${\cal S}$ and the topology of the power--law random graphs $\mathbb{G}_{\alpha,\gamma}(n,k)$ is found to exist. Indeed, for $2.3 < \gamma < 4.5$ the Euler--Poincar\'e characteristic $\chi(\mathbb{G}_{\alpha,\gamma}(n,k))$ is supplied only by $\beta_0$ and $\beta_1$ as follows from Fig. \ref{SF} and Eq. \eqref{E-P}.
\begin{figure}[ht]\centering
\includegraphics[scale=.50]{SF}
\caption[10 pt]{Behaviours of the entropy ${\cal S}/n$ and Betti numbers $\beta_0$, $\beta_1$ of power-law $\mathbb{G}_{0,\gamma}(200,k)$ networks as a function of the exponent $\gamma$; $k$ values varied - according
to the realization of the RG and, independently, of $\gamma$ - approximately in the range $160-500$.}\label{SF}
\end{figure}
Finally, in both figures, Fig. \ref{RG} and Fig. \ref{SF}, the pattern of ${\cal S}$ shows a sort of saturation of its values. The reason of this relies on the numerical methods for regularizing the volume computed through the volume form \eqref{volumeform}. Interestingly, the apical values of $\mathbb{G}_{\alpha,\gamma}(n,k)$ and $\mathbb{G}(n,k)$ are different thus allowing to characterise different network ensembles via the geometric--entropy ${\cal S}$ defined in Eq. \eqref{entropy}.
\section{Concluding remarks}\label{sec5}
In this work we have pursued our investigation about the potential applications of a recently defined geometric--entropy which has been shown to perform very well in quantifying networks complexity \cite{FMP,Franzosi15,Franzosi16}. The present investigation focused on the possible use of the mentioned geometric entropy to catch some topological property of networks. By considering networks as clique graphs, we proceeded in a bottom--up analysis. To begin with, small--size networks have been considered and analytical computations have been applied. Then, numerical computations has been used to tackle large--size networks.
The entropy of a network is obtained after having associated to it - on the basis of a probabilistic approach - a Riemannian manifold, and then by computing the volume of this manifold. Since in general this manifold is not compact, we introduced an ``infra--red'' and ``ultra--violet'' regularising function to compactify it. This procedure is independent of networks topology in as much as it is defined up to graph isomorphisms.
Small--size networks are ascribed to a $4$--dimensional simplicial complex $K$. The analytical computation of ${\cal S}$ for these networks displays a monotonic behaviour of ${\cal S}$ with respect to the Betti numbers $\beta_0$ and $\beta_1$. However, the information about network complexity retained by the geometric entropy cannot be reduced to the only knowledge of the topological properties described by $H_0$ and $H_1$. This is due to the fact that $\beta_0$ and $\beta_1$ do not exhaustively account for the network connectivity. This explains why different values of ${\cal S}$ are found for networks with same $\beta_0$ or $\beta_1$ (see Tab. \ref{Tab3}). Then, passing to a ``coarse--grained'' description for large networks, the connection between ${\cal S}$ and topology becomes clearer.
We have considered two different network ensembles in tackling large--size graphs, that is, the random graphs and the scale--free networks. The entropy of random graphs perfectly correlates with their $\beta_0$. This is not very surprising because $\beta_0$ counts the number of connected components of a graph. Thus, due to the appearance of a giant component in random graphs, $\beta_0$ is expected to asymptotically reach the value $1$, as well ${\cal S}$ saturates for large values of $k$. As far as $\beta_1$ is concerned, we found a proper correlation between the entropy of scale--free networks and their number of cycles. Indeed, scale--free models account for networks with a finite numbers of cycles and the behaviour of ${\cal S}$ properly agrees with the pattern displayed by the Betti number $\beta_1$ of the considered power--law random graphs. This suggests a strong correlation between ${\cal S}$ and the topology of scale--free networks.
Finally, from both Figure \ref{RG} and Figure \ref{SF}, we can see that the entropy ${\cal S}$ saturates for some values of $k$ and $\gamma$, respectively. Since these apical values are different, it seems that the geometric--entropy ${\cal S}$ is able to characterise some difference within the network ensemble. This paves the way to further and deeper studies also about this issue.
\begin{acknowledgments}
We are indebted with M. Piangerelli, M. Quadrini, and V. Cipriani of the Computer Science Division of the University of Camerino for computational help. D.F. also thanks E. Andreotti for useful discussions.
\end{acknowledgments}
| 40,256
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TITLE: Restriction of rational functions to closed subvarieties
QUESTION [2 upvotes]: I'm confused about a basic point regarding the definition of the ring of regular functions on a closed subvariety. Let $X=\text{Spec}(A)$ be an affine variety (I'm thinking of varieties as integral, separated schemes of finite over an algebraically closed field but I don't think it matters here). Let $\mathfrak{p}$ be a prime ideal in A; we have a closed subscheme $Y=\text{Spec}(A/\mathfrak{p})\cong V(\mathfrak{p})\subset X$. Therefore the regular functions on $Y$ are the restrictions of regular functions on $X$, given by the quotient map $A\to A/\mathfrak{p}$. In particular I'm thinking about $A = \mathbb{C}[x,y]$, with $\mathfrak{p} = (f)$ for some irreducible polynomial $f$, so the regular functions on $V(f)$ are just polynomials in $x$ and $y$, identified if they agree on $V(f)$.
However, on an open set $U\subset X$, there are additional regular functions other than the restrictions of regular functions on $X$; in particular, we allow rational functions with poles outside $U$. For concreteness, let $h\in A$ be irreducible and suppose that $U=D(h)$, so we have the distinguished inclusion $U\cong \text{Spec}(A_h)\hookrightarrow \text{Spec}(A)$. The regular functions on $U$ then look like $\frac{g(x,y)}{h(x,y)^n}$ for $g\in A$ and $n\geq 0$.
Question:
So what happens if $Y\subset U \subset X$? (That is, $V(f)\cap V(h)=\emptyset$.) Thinking of $Y$ as a closed subvariety of $U$, its regular functions are restrictions of regular functions on $U$ (and thus rational functions on $X$), so they are elements of $A_h/(f)$. It's not clear to me how to show in general (if it's even true!) that $A_h/(f)\cong A/(f)$, and so these are in fact the same scheme structure on $Y$.
I suspect that these should in fact be the same structure, and so any rational function like $\frac{g(x,y)}{h(x,y)^n}$ restricted to $V(f)$ should be the same as the restriction of some polynomial $q(x,y)$ to $V(f)$ (again, assuming that $h$ vanishes nowhere on $V(f)$). The only cases that I can come up in $\mathbb{A}_{\mathbb{C}}^2$ of two nonintersecting hypersurfaces are two parallel lines, and there it is clear that, if $f=0$ and $h=0$ define these lines, then $h$ is constant on $V(f)$ so these rational functions agree with polynomials. But I have no idea how one would deal with this in general cases where the intersections or lack thereof are much less obvious.
REPLY [2 votes]: These two rings are in fact the same, so we can consider the subvariety $Y$ directly in $X$, or $Y$ in $U$ in $X$, and and the resulting structure sheaf is the same. Note that $h$ is already invertible in $A/(f)$: since $h$ doesn't vanish at any point of $V(f)$, this mean it isn't contained in any maximal ideal of $A/(f)$, so it must be a unit. Therefore $(A/(f))_h = A/(f)$. Now we use the fact that localization commutes with quotients, so $A_h/(f)\cong (A/(f))_h$. Putting these facts together, we have $A_h/(f)\cong A/(f)$. Therefore the regular functions are in fact the same.
| 64,735
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| 182,737
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\begin{document}
\maketitle
\begin{abstract}
\input{abstract}
\textbf{Keywords}: Taylor state, stellarator, generalized Debye
sources, plasma equilibria, Laplace-Beltrami
\end{abstract}
\section{Introduction}
The computation of magnetohydrodynamic (MHD) equilibria in toroidal
domains without axisymmetry is a notoriously challenging problem,
having both computational roadblocks as well as touching on
subtle mathematical questions~\cite{grad67,Bruno1996,Hudson2010,HelanderReview}. Until
recently, computational efforts to solve this problem could be divided
in two categories~\cite{Harafuji1989}. The first category of numerical
solvers relies on the assumption of the existence of nested magnetic
flux surfaces throughout the computational
domain~\cite{Bauer1978,Hirshman_VMEC1,Hirshman_VMEC2,Hirshman_VMEC3,Taylor1994}.
These solvers have played an important role in the design of new
non-axisymmetric magnetic confinement devices as well as the analysis
of experimental results obtained from existing ones. However, they are
fundamentally limited in terms of both robustness and accuracy for the
computation of equilibria with both a smooth plasma pressure profile
and smooth magnetic field line pitch. In this regime, this class of
solvers (and the model upon which they are based) is unable to
accurately approximate the singular structures in the current density
which must naturally occur in such
situations~\cite{HelanderReview,Loizu2015,ReimanComparison,Loizu2016}.
On the other hand, an alternative, second class of solvers, does not constrain the
space of solutions to equilibria with nested flux surfaces, and
computes equilibria which may have magnetic islands and chaotic
magnetic field
lines~\cite{Reiman1988,Harafuji1989,Suzuki_Hint2}. These solvers also
play an important role in the magnetic fusion program since they can
be used to study, among other significant questions, the disappearance
of magnetic islands, often called \emph{island healing}, corresponding
to an increase of the plasma pressure~\cite{Hayashi1994,Kanno2005} or
to a change of the coil configuration~\cite{Hudson2002}. They are also
often able to compute the details of the magnetic field configuration
in the vicinity of the plasma edge~\cite{Drevlak2005}. Despite these
additional capabilities, equilibrium codes in the first category are
often favored because existing solvers in the second category converge
substantially slower~\cite{Drevlak2005} and are much more
computationally intensive~\cite{Hudson2007}.
\input{highlight}
Recently, a third approach has been developed which combines aspects
of the two categories of solvers described above. In this approach,
the entire computational domain is subdivided into separate regions,
each with constant pressure, assumed to have undergone Taylor
relaxation~\cite{Taylor1974} to a minimum energy state subject to
conserved fluxes and magnetic helicity. Each of these regions is
assumed to be separated by ideal MHD
barriers~\cite{Hudson2012,HudsonPPCF2012}. This model has a
significant limitation: for general pressure profiles, solutions of
this model only truly agree with the equations of ideal MHD
equilibrium when one takes the limit of infinitely many
interfaces~\cite{Dennis2013}. When the number of interfaces is
finite, the model is only a first-order approximation of the ideal MHD
equilibrium equations, independent of any subsequent numerical
discretization. This third approach is nevertheless very promising
for several reasons. First, unlike the more general ideal MHD
equilibrium case, existence of solutions for the stepped pressure
equilibrium model has been established for tori whose departure from
axisymmetry is sufficiently small~\cite{Bruno1996}. Second, numerical
solvers for this model can be used to study equilibrium configurations
with magnetic islands and regions with chaotic magnetic field
lines~\cite{Loizu2017} for a computational cost which is substantially
smaller than that of ideal MHD solvers having similar
capabilities~\cite{Reiman1988,Harafuji1989,Suzuki_Hint2}. Furthermore,
they generally have more robust convergence properties. Finally, the
model is well-suited for rigorous error convergence analysis and code
verification~\cite{Hudson2012,Loizu2016Verification}.
At present time, the only equilibrium code in the third category is
known as the Stepped Pressure Equilibrium Code
(SPEC)~\cite{spec-code}. For a given plasma pressure profile,
the computation of equilibria using SPEC is an iterative process. On
each iteration, one first computes the magnetic field~$\vector{B}$ of
the Taylor states inside each region, given by
~$\nabla\times\vector{B}=\lambda\vector{B}$ with~$\lambda$ a
specified constant (possibly different in each region). Then, the
force-balance condition on the ideal MHD interfaces is verified. If
the total pressure $p+|\vct{B}|^2/2$, where $p$ is the plasma
pressure, is continuous across each interface to the desired numerical
precision, the iterative process stops. If it is not continuous to the
desired numerical precision, the shape of the MHD interfaces is
modified in order to satisfy force balance, and a new iteration
starts. This shape optimization is a highly nonlinear procedure.
In this article, we focus on the first step within each iteration,
namely the computation of Taylor states given by
$\nabla\times\vector{B}=\lambda\vector{B}$ in toroidal domains for
which the boundary is given, and~$\lambda$ and the flux conditions are
such that the problem is well-posed~\cite{O_Neil_2018_Taylor}. In
SPEC, this is done using a Galerkin approach~\cite{Hudson2012} in
which one solves for the magnetic vector potential~$\vector{A}$. The
components of~\vct{A} are represented using a Fourier-Chebyshev
expansion; the Fourier representation captures the double periodicity
in the poloidal and toroidal angles and the Chebyshev representation
is used for the radial variable~\cite{Loizu2016Verification}. In
contrast, in this work we present a boundary integral representation
for the Taylor state~$\vector{B}$ based on a single scalar variable
and solve the associated boundary integral equation. The advantages of
our numerical method as compared to the solver in SPEC are the typical
advantages one expects from integral equation solvers: the number of
unknowns in our approach is much smaller than in SPEC since unknowns
are only needed on the surface of the domain, our solver avoids issues
with the coordinate singularity which occurs when parameterizing the
volume of genus-one domains~\cite{HudsonPPCF2012}, and the
representation immediately leads to a well-conditioned (away from
physical interior resonances) second-kind integral equation which can
be numerically inverted to high-precision. We will present numerical
tests which demonstrate the robustness and efficiency in our approach,
showing that for a given target accuracy, our solver is significantly
faster than the SPEC code and that it avoids conditioning issues
encountered in SPEC.
Our integral equation formulation is based on the same generalized
Debye representation that we presented
in~\cite{O_Neil_2018_Taylor} for the calculation of axisymmetric
Taylor states. One may initially think that generalizing the solver
from axisymmetric geometries to non-axisymmetric ones is
straightforward. This is not so, for several reasons. First, the
numerical discretization of the boundary integral equation formulation
requires quadratures for weakly-singular kernels on the boundary of
the domain. In axisymmetric geometries, the boundary of the domain can
merely be considered a closed curve and fast and accurate quadrature
schemes are readily available (and easy to implement). However, for
the computation of Taylor states in non-axisymmetric geometries, the
boundary is a surface; this makes the accurate and fast computation of
these integrals much more challenging, both from a mathematical and
computational point of view. The same difficulties apply to
the computation of the surface gradient and the inverse
Laplace-Beltrami operator, which are much easier to evaluate along a
closed curve than on general stellarator geometries. Finally, in
axisymmetric domains, there exists simple closed form expressions for
the surface harmonic vector fields required in our formulation, and
this is not the case in non-axisymmetric domains. A significant
portion of this article focuses on the new algorithms we developed to
address these difficulties. We have implemented our method in the form
of a software library called BIEST (Boundary Integral Equation Solver
for Taylor states). The library is made publicly available for use by
the scientific community and to allow independent verification of
our results\footnote{\url{https://github.com/dmalhotra/BIEST}}.
The paper is organized as follows. In Section~\ref{sec:debye}, we
present a brief review of the generalized Debye representation for
magnetic fields satisfying the Taylor state equation, and of the
resulting boundary integral formulation for the computation of the
Taylor state. We also give a new numerically stable representation for
the computation of the flux condition when $\lambda$ approaches zero,
and present an alternative, more direct method for computing vacuum
fields (i.e. the case where $\lambda=0$). In Section~\ref{s:algo} we
provide a detailed description of the numerical solver,
with a particular emphasis on high-order surface
quadratures for singular kernels and inverting the Laplace-Beltrami
operator. In Section~\ref{s:tests}, we test the accuracy and speed of
the singular quadrature scheme and of the Laplace-Beltrami solver, as
well as the accuracy and speed of the entire solver. In particular, we
compare the performance of our solver with that of SPEC for the
W7-X geometry~\cite{Beidler1990,Pedersen2017}, a stellarator
experiment in Greifswald, Germany. We summarize our work in
Section~\ref{s:summary} and propose directions for future work.
\section{Beltrami fields, Taylor states, and generalized Debye
sources}\label{sec:debye}
In this section we detail the relationship between time harmonic
electromagnetic fields and Taylor states. These two classes of vector
fields can be directly linked using the generalized Debye integral
representation for Taylor states. Using this integral representation,
second-kind boundary integral equations are then derived for
force-free fields corresponding to Taylor states in stellarator
geometries. These boundary integral equations can then be used to
solve for stepped-pressure equilibria, as discussed in the
introduction.
As discussed in the introduction, in this article we focus solely on
the task of computing Taylor states in stellarator geometries.
Taylor states are
described by the equation
\begin{equation}
\nabla \times \vector{B} = \lambda \vector{B}
\label{eq:TaylorState}
\end{equation}
where~$\lambda$ is a given constant throughout the computational
domain, determined based on magnetic energy and helicity.
As is well-known~\cite{Hudson2012,O_Neil_2018_Taylor}, this
partial differential equation needs boundary conditions and flux
constraints on $\vector{B}$ in multiply-connected geometries
in order to be well-posed. We will specify
them shortly, but refrain from doing so at this stage in order to
focus on the generalized Debye representation for~$\vector{B}$ which
\emph{a priori} satisfies~\eqref{eq:TaylorState}, independent of the
boundary conditions or the flux constraints.
Since~$\lambda$ is a real-valued constant, it is easy to
see that by setting~$\vector{E} = i\vct{B}$ the pair~$\vct{E},
\vct{B}$ satisfy the source-free time-harmonic Maxwell equations with
wavenumber~$\lambda$, denoted by THME$(\lambda)$:
\begin{equation}\label{eq:thme}
\begin{aligned}
\nabla \times \vct{E} &= i\lambda \vct{B}, \qquad&
\nabla \times \vct{B} &= -i\lambda \vct{E}, \\
\nabla \cdot \vct{E} &= 0 , &
\nabla \cdot \vct{B} &= 0.
\end{aligned}
\end{equation}
There is a rich literature on integral equation
representations and methods for solving various boundary value
problems for the THME$(\lambda)$. It turns out that the generalized
Debye source representation for solutions to the THME is also
particularly well-suited for solving boundary value problems for Taylor states~\cite{epstein2015}. We now turn to
a very brief overview of this representation, and the derivation of an
integral representation and integral equation for Taylor states in
magnetically confined plasmas. Various theoretical and numerical
aspects of the generalized Debye source representation can be found
in~\cite{Epstein_2012,epstein2015,O_Neil_2018_Taylor,Epstein_2010,epstein2019}.
\subsection{Generalized Debye sources}
\input{overview}
\subsection{Taylor state formulation\label{ss:formulation}}
\input{formulation}
\section{A fully 3D Taylor-state solver \label{s:algo}}
\input{algo}
\section{Numerical experiments}\label{s:tests}
\input{results}
\section{Summary}\label{s:summary}
\input{conclusions}
\appendix
\section{Notation}
In \pr{t:notation}, we list some frequently used symbols for easy
reference.
\input{notation-table}
\bibliographystyle{abbrvnat}
\bibliography{ref}
\end{document}
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\section{Case study: US Interstate 210}
\label{sec_simulate}
This section presents the design of ramp metering for the I-210E stretch that we introduced in Section I. Concretely, we applied the localized and partially coordinated control design based on the demand data and the calibrated SS-CTM. Although the proposed methods have been already illustrated with using numerical examples, this case study aims at evaluating and comparing them under more realistic conditions, in particular time-varying demand.
We consider five scenarios. In the first scenario there is no ramp metering, and the remaining four adopt different control strategies. Recall from Fig.~\ref{fig_calibratedSpeedMap} that the traffic congestion occurred between 13:00 and 19:00; so we considered ramp metering turned on in that period. The second and the third scenarios correspond to the localized ALINEA and the coordinated METALINE, respectively. The METALINE controller is given by
\begin{equation}
\mu(t) = \mu(t-1) - K_P(n(t) -n(t-1)) - K_I(n(t)-n^c), \label{eq_metaline}
\end{equation}
where $K_P$ and $K_I$ are gain matrices, and $n^c\in\mathbb{R}^K$ are the nominal critical densities. We computed the control parameters $K_P$ and $K_I$ by solving a mixed-integer bilinear program that minimizes \emph{vehicle hours traveled} $H$ for the nominal CTM:
\begin{equation}
H = \Delta_t \Big(\sum_{k=1}^K \sum_{t=1}^T l_k n_k(t) + \sum_{k=1}^{K} \sum_{t=1}^T q_k(t) \Big), \label{eq_vht}
\end{equation}
where $\Delta_t$ denotes time step size. The optimal VHT between 13:00 and 19:00 attained 9829 veh$\cdot$hr under METALINE. In the last two scenarios, the localized and partially coordinated ramp metering were designed for each hour. We first computed hourly mainline flows, on-ramp flows and mainline ratios along the highway (see Fig.~\ref{fig_hourly}) and then used these hourly values for computing the control parameters.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\linewidth]{Images/LongHill.eps}
\caption{Hourly traffic demands around Long Hill Avenue.}
\label{fig_hourly}
\end{figure}
The four ramp metering strategies were evaluated via numerical simulation. For SS-CTM the highway traffic was simulated from 12:00 to 21:00 with a time step size $\Delta_t=10$ seconds and each control strategy was tested for 1000 samples.
While implementing ramp control, we also considered the capacity limits of on-ramp queues. In practice, ramp metering can result in long queues that spill from on-ramps and block street traffic. To avoid this, we restricted the maximal queue size at 40 veh/lane. Once the simulated queue size exceeded the threshold, the ramp metering would stop and on-ramp vehicles would enter the mainline at the maximum rate.
We consider two performance metrics, namely 1) time-averaged buffer queue length
$$\hat{Q}=\frac{1}{T}\sum_{t=1}^T\sum_{k=1}^K q_k(t)$$
and 2) vehicle hours traveled \eqref{eq_vht}. The first is the control objective of the original control problem $\mathrm{P}_0$, and the second is commonly used for measuring the efficiency of highway operation, which includes mainline traveling time and queuing time of all vehicles. Tables~\ref{tab_compareQ}-\ref{tab_compareVHT} summarize the evaluation of control strategies.
Though the performance of our control design approach is marginal in the initial hours, they are effective in significantly reducing the queue length between 15:00-19:00. For example, compared with ALINEA (resp. METALINE), our localized control (resp. partially coordinated control) can shorten the queue length by 12.8\% (resp. 8.8\%) during 15:00-19:00. Overall, the localized and partially coordinated control designs reduce vehicle hours traveled by 8.3\% and 9.9\% respectively, while the classical ALINEA and METALINE decrease VHT by 5.1 \% and 6.2\%. This demonstrates that the highway system performance can be improved by considering stochastic capacity in designing ramp controllers. In this paper, we considered the affine controller \eqref{eq_controller} that relies on local measurements. Potentially higher gains can be achieved by considering more sophisticated control strategies based on more upstream or downstream measurements.
\begin{table}[htbp]
\centering
\scriptsize
\caption{Time-averaged queue length (veh).}
\begin{tabu}to\linewidth{X[4.3,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1.1,c,m]}
\toprule
& 13:00-14:00 & 14:00-15:00 & 15:00-16:00 & 16:00-17:00 & 17:00-18:00 & 18:00-19:00 & Mean \\
\midrule
No control & 545 & 802 & 771 & 856 & 1087 & 1148 & 868 \\
ALINEA & 548 & 781 & 862 & 979 & 1041 & 909 & 853 \\
METALINE & 555 & 741 & 811 & 909 & 983 & 875 & 812 \\
Localized & 555 & 722 & 747 & 855 & 896 & 802 & 777 \\
Partially coordinated & 555 & 722 & 764 & 855 & 900 & 742 & 756 \\
\bottomrule
\end{tabu}
\label{tab_compareQ}
\end{table}
\begin{table}[htbp]
\centering
\scriptsize
\caption{Vehicle hours traveled (veh$\cdot$hr).}
\begin{tabu}to\linewidth{X[4.3,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1,c,m]X[1.1,c,m]}
\toprule
& 13:00-14:00 & 14:00-15:00 & 15:00-16:00 & 16:00-17:00 & 17:00-18:00 & 18:00-19:00 & Sum \\
\midrule
No control & 1795 & 2177 & 2476 & 2862 & 3040 & 2578 & 14928 \\
ALINEA & 1770 & 2112 & 2382 & 2719 & 2816 & 2366 & 14165 \\
METALINE & 1764 & 2099 & 2348 & 2658 & 2787 & 2350 & 14006 \\
Localized & 1761 & 2078 & 2320 & 2592 & 2691 & 2247 & 13689 \\
Partially coordinated & 1760 & 2078 & 2282 & 2511 & 2619 & 2201 & 13451 \\
\bottomrule
\end{tabu}
\label{tab_compareVHT}
\end{table}
Fig.~\ref{fig_case_denmap} visualizes the difference between traffic density maps with and without ramp metering. Each subfigure presents a density map under a particular control strategy minus the background density map without ramp control (see Fig.~2(c)). The positive values indicate that the traffic jam is alleviated by the ramp metering and the negative values imply that the congestion worsens. We note that ALINEA significantly mitigates the traffic jam around Lone Hill Avenue. Our localized ramp control further reduces the congestion upstream of Irwindale Avenue. This can be ascribed to the controller around Barranca Avenue that is aware of the capacity variation. Similarly, METALINE also reduces the upstream congestion by coordinated control, but our capacity-aware partial coordination further ameliorates congestion around Lone Hill Avenue and upstream of Barranca Avenue.
| 26,075
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TITLE: the set $X '$ of limit points of $X$ is compact
QUESTION [3 upvotes]: Let $(M,d)$ be a metric space. If for each $\emptyset\neq A, B ⊂ M$ closed disjoint, we have
$d (A, B) =\inf \{d (x, y): (x, y) ∈ A \times B\}> 0$ then there exists $K ⊂ M$ compact such that for any neighborhood $V$ of $K$, the set $M \setminus V$ is is uniformly discrete (i.e, exists $\delta >0$ such that $d (x, y) \geq \delta$ for any $x, y \in M$, with $x\neq y$).
My idea is to show that $K$ is the set of accumulation points of $M$. I observed that if $K$ is the set of accumulation points the result is already there. Indeed, if $ M \setminus K $ is not uniformly discrete, there exist $x, y \in M\setminus K$ with $x\neq y$ such that $d (x, y) <\delta$ for all $\delta> 0$ then $y \in B(x,\delta)\setminus\{x\} \cap M$, i.e $x$ is an accumulation point, which is absurd.
under the hypothesis that $d (A, B) =\inf \{d (x, y): (x, y) ∈ A \times B\}> 0$, is it true that the set of accumulation points is compact?. I have tried to demonstrate it by contradiction but I cannot find the absurdity. I appreciate the help.
REPLY [2 votes]: $\newcommand{\cl}{\operatorname{cl}}$Here’s a slightly different (though very similar) approach that I’m posting mostly so that I can easily find it again.
Let $D$ be a set of isolated points of $M$ that is not uniformly discrete. Clearly no cofinite subset of $D$ is uniformly discrete, so we can recursively define sequences $\langle x_n:n\in\Bbb N\rangle$ and $\langle y_n:n\in\Bbb N\rangle$ in $D$ such that the points $x_n$ and $y_n$ are all distinct, and $d(x_n,y_n)<2^{-n}$ for each $n\in\Bbb N$. Let $H=\{x_n:n\in\Bbb N\}$ and $K=\{y_n:n\in\Bbb N\}$; then $H$ and $K$ are disjoint, but $d(H,K)=0$, so at least one of them has a limit point $p\in M'$, the set of non-isolated points of $M$. In fact it’s clear that $p\in(\cl H)\cap\cl K$, but all that we really need is that $p\in\cl D$.
Now let $U$ be an open nbhd of $M'$, and suppose that $D\subseteq M\setminus U$ is not uniformly discrete. It follows from the previous paragraph that there is some $p\in M'\cap\cl D\subseteq U$ and hence that $U\cap D\ne\varnothing$, contradicting the choice of $D$. Thus, $M\setminus U$ is uniformly discrete, and it only remains to show that $M'$ is compact.
If $M'$ is not compact, there is a countably infinite set $D=\{x_n:n\in\Bbb N\}\subseteq M'$ that has no limit points in $M'$. $D$ is a closed discrete subset of $M$, so for each $n\in\Bbb N$ there is an $r_n>0$ such that $B(x_n,r_n)\cap D=\{x_n\}$, and we may assume that $r_n<2^{-n}$. Finally, since $x_n\in M'$, there is a $y_n\in B(x_n,r_n)\setminus\{x_n\}$. Let $E=\{y_n:n\in\Bbb N\}$ and argue much as in the first paragraph: $D\cap E=\varnothing$, $d(D,E)=0$, and $D$ is closed, so $E$ is not closed. Let $p\in(\cl E)\setminus E$; clearly $p\notin D$, so there is an $\epsilon>0$ such that $B(p,2\epsilon)\cap D=\varnothing$. $B(p,\epsilon)\cap E$, however, is infinite, so there is an $n\in\Bbb N$ such that $y_n\in B(p,\epsilon)$ and $2^{-n}<\epsilon$ and hence
$$d(p,x_n)\le d(p,y_n)+d(y_n,x_n)<2\epsilon\,,$$
which is impossible. Thus, $M'$ contains no such set $D$ and is therefore compact.
| 137,806
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BUP Library and Archive started with approx 500 books in 2010 with a room space of 200 square-feet in the 3rd floor of MIST Administrative Building. At present the library is fully air-conditioned with a floor space of approx 10600 square-feet, equipped with all learning amenities and is striving to utilize cutting-edge technology as well as to incorporate world famous electronic scholarly assets. As on September 2019 the library has a collection of mostly department related 17000 books and 2800 journals/magazines. The library users are able to do check-out and check-in through RFID based library management system. Besides having a cyber corner users are also provided with off-campus facility with RemoteXs to browse 12 international E-resource links.
Useful Links
BUP Library Management System
E-Subscriptions
Cambridge University Press
| 112,036
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TITLE: Evaluate: $\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx$
QUESTION [3 upvotes]: How can I evaluate: $$\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx$$
How I proceed:
$$\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx=2\lim_{h \to 0} \frac{1}{h}\int_{0}^{1}\frac{1}{1+(\frac{x}{h})^2}~dx=2\lim_{h \to 0}\frac{1}{h}\arctan\frac{1}{h}$$
Then how can I prooceed. Please help. Thank in advance.
REPLY [1 votes]: $$\lim_{h \to 0} \int_{-1}^{1}\frac{h}{h^2+x^2}~dx=2\lim_{h \to 0} \frac{1}{h}\int_{0}^{1}\frac{1}{1+(\frac{x}{h})^2}~dx=2\lim_{h \to 0} \int_{0}^{1/h}\frac{1}{1+(\frac{x}{h})^2}~d\left({x\over h}\right) $$
$$ 2\lim_{x\rightarrow 0} \arctan(1/x) = \pi \text{ as stated above.}$$
| 157,625
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Broxxx 671 Report post Posted February 8 Outburst massed up 30 washed up Panthers for one night only for a F2P Prep vs Divine. Despite not playing the game for almost a year we managed to come away with the win. Thanks for the fight dudes I cba posting r1,r2,r3 etc but we won so enjoy the povs ty xo Andrew POV @ZigPOV Share this post Link to post Share on other sites
| 304,936
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TITLE: How to show Let $(X, d)$ be a metric space. Prove that if $X$ has a dense finite subset, then $X$ itself is finite?
QUESTION [1 upvotes]: Let $(X, d)$ be a metric space. Prove that if $X$ has a dense finite
subset, then $X$ itself is finite.
My attempt: if possible, $X$ is infinite and $A$ be a finite dense subset of $X$. Now for any point $x$ in $X$,any neighborhood of $x$ contains at least one point from $A$. Since $X$ contains infinite number if points and $A$ is finite there will be some contradiction which is not coming in my mind. Please I need help.
REPLY [3 votes]: You aren't far off. If $A$ is a subset of the metric space $X$ that is dense and finite, then $A = X$. To see this assume (for a contradiction) that $A$ a proper finite dense subset of the metric space $X$ and let $x \in X \setminus A$. As $x \not\in A$, $d(x, a) > 0$ for every $a \in A$. But then if we take $\varepsilon = \frac{\min_{a\in A}d(x, a)}{2}$ (which is well-defined and positive because $A$ is finite), there is no element $a \in A$ with $d(x, a) < \varepsilon$, so $A$ is not dense.
| 213,626
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I was recently requested to help a non-profit organization develop a public relations action plan they are able to execute without the employment of a PR firm by themselves.
Here’s what we did.
There are 2 distinct stages of developing a public relations action plan
Placing and
Carrying out the strategy
PHASE 1 – PLACING THE BASIS
There are 7 steps to building the base to your public relations action plan. Don’t bypass any measure or take action just midway. All of the leaders of the organization need to participate in this stage in order to come to an unambiguous basis to build on. I propose a meeting dedicated to that one job.
1. Who Are You Talking To?
First, identify who your end crowd is, that’s to say who the press is going to be speaking to on your own behalf. This can help guide the manner in which your message is personalized by you. In the instance of the non-profit organization that I developed the strategy for it was twofold: (a) the present members and (b) the members of the public. The messages they send out to both groups is not always the same.
2. What’s Your Message?
Create the message that you just are striving to disseminate. I know it seems counter intuitive however do not talk about your list of nonprofits organization, talk about what itdoes for the community. Your message should serve the requirements of your crowd… not your non-profit organization.
| 137,113
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Last week, I wrote a post about the differences between content types, taxonomies, and entities in Drupal, and how you should use each. Now, even with having a strong understanding of how to use these mechanisms, there is still plenty of room to mess up when developing a content model. Here are ten lessons we have learned over the past seven years of building Drupal-powered websites and apps:
1. Using content types to represent content assets
A very common mistake made in Drupal is using a content type to represent things like photos, videos, etc. that are then associated with other pieces of content. This stems from the poor support for media management that has been available within the system. Developers had to give direct access to a file directory on the server for website administrators to upload and manage images and video, which was less than elegant.
The alternative was to create a content type for photos, videos, etc., and use node reference fields to allow users to add those assets to a node. The downside of that is you now have hundreds, if not thousands, of individual nodes that represent what should be represented within a field on a content type.
With the introduction of entities in Drupal, and the release of the Media module, media management in Drupal has gotten better. We default to using the Media module because it removes all the clutter of having nodes that represent assets within the content administration screen, and creates a nice interface where website administrators can view, edit, and delete media files.
2. Using taxonomy to represent content
Taxonomy vocabularies are meant to categorize content, not represent content. This seems likes a no-brainer, but we have come across many websites that use taxonomy as a way to create content types.
A good rule of thumb is if the terms in your vocabulary are not repeatedly used across multiple pieces of content, or if the metadata association with your vocabulary terms is what you hope for your readers to engage with more than the term itself, it should probably be a content type.
3. Using a content type to organize other content types
Unless you are creating a hierarchy to your content (such as including articles within a newsletter issue), content types shouldn’t be used to organize content. Taxonomy vocabularies are fieldable if you need to associate any metadata with vocabulary terms. We, as much as possible, try to leverage taxonomies over content types so we can make the workflow for content editors more streamlined.
4. Creating new fields to represent the same type of data as existing fields
Drupal creates a custom multiple tables in your database for each field that you create on a content type. Your database can quickly bloat, and queries for content can begin to impact performance.
Drupal allows you to reuse fields that you have already created in the system on multiple content types. As an example, if you are adding a thumbnail image to a Blog Post and Event content types, then you can easily reuse the same field for each. This will save you some performance hits and time in building out your content types.
5. Creating too many content types
It is easy to fall into the trap of creating a content type for everything on your website. We have seen websites with upwards of twenty content types, with some representing the same type of content.
Think very broadly when creating content types. You may not need two blog content types, one for your rapid response blog versus one for your policy blog. Even if these two types of posts have slightly different data models, think of you you can leverage conditional fields or taxonomies to categorize this type of content so that it will give the editor the fields they need to produce content and allow you to present that content in unique places across your website.
6. Creating content types content creators/editors don’t have the capacity to produce
It is easy to listen to your clients and create the content types that they ask you to create. Seriously ask yourself “does my client have the capacity/resources to develop the type of content they are asking for?” Most times, the answer is no.
Content management systems are like self-serve ice cream bars. You think you can handle the six large scoops with four different toppings until you take that third bite. The same thing with content types - clients can tend to want all the content types they think they will produce, but the reality is they won’t.
Take care to really understand the ways your clients product content and what they can realistically produce to meet their desired goals. Also, perform research into what content their audience(s) really want and have the tough conversation with your client about eliminating any content types that aren’t necessary to meet their users’ needs.
7. Allowing website administrators to format content within the WYSIWYG editor when they don’t need to
I won’t dive into this too much to avoid the content blobs versus chunks debate in our comment stream. Looking at this from a sheer editorial experience standpoint, it can be easier for content creators to populate fields rather than having to format content within a WYSIWYG field.
8. Using fields to manage presentation of content
Fields on the content types should not be used to manage how content is presented to the end user. I have seen fields used to assign classes to content fields; assign nodes to regions of a page; and even ordering nodes within a listing. There may be some edge cases where doing this makes sense, but it is more appropriate to leverage the tools within Drupal to manage presentation of content, such as Panels, Views, Nodequeue, etc.
These are just some of the most common pitfalls that were top of mind for our team. Are there other pitfalls you have found when modelling content within Drupal?
| 268,518
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TITLE: What is a quantum simulator?
QUESTION [0 upvotes]: What is the idea behind of quantum simulator aimed to study properties of matter, such as using quantum dots to study the exotic quantum states?
REPLY [2 votes]: Although we typically think of the laws of quantum mechanics as governing phenomena only at the smallest length scales, there are situations in which we can measure quantum effects at reasonable experimental sizes. Superconductivity is a great example of this insofar as a macroscopic piece of superconducting material can have a quantum coherence length of the same scale as itself. Crystalline nanoparticles, such as those of CdSe which are grown with little difficulty in chemistry laboratories across the world, are another great example since their macroscopic properties (e.g., color) are intimately tied up with quantum effects due to their small size. These are not simulators, per se, but systems in which quantum effects manifest themselves at a human scale.
True quantum simulators are experimental systems which are constructed to follow the same mathematical laws as a particular quantum phenomenon that defies (or at least frustrates) experimental investigation. These simulators display non-classical behavior themselves, but can be modified so that their time evolution mimics that of some other quantum system. Examples include optical quantum simualtors and the quantum dots you describe. As noted by Richard Feynman, such a simualtor can act as a quantum computer and (as was determined later) vice versa.
| 187,272
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TITLE: Where can I find a proof of ($\aleph_1 \leq 2^{\aleph_0}$ is independent of ZF)?
QUESTION [8 upvotes]: In "A tutorial on countable ordinals" [1], in page 25, Forster uses the fact that $\aleph_1 \leq 2^{\aleph_0}$ is independent of ZF to prove that there is no definable family of fundamental sequences up to $\omega_1$. I know the proof must be technical, and I can't find anywhere a proof of that. Does anyone know of any book or paper that proves it? Thanks!
[1] https://www.dpmms.cam.ac.uk/~tf/fundamentalsequence.pdf
REPLY [5 votes]: The easiest places to find this consistency proof are the following papers:
Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable, Ann. Math. (2) 92, 1-56 (1970). ZBL0207.00905.
Truss, John, Models of set theory containing many perfect sets, Ann. Math. Logic 7, 197-219 (1974). ZBL0302.02024.
Some remarks:
If $\aleph_1\nleq2^{\aleph_0}$, then there is no real number $x$ such that $L[x]$ computes $\omega_1$ correctly. Therefore $\omega_1$ is a limit cardinal in $L$.
If you assume countable choice holds for sets of real numbers, then $\omega_1$ is regular, and by the above, it is inaccessible in $L$. If you don't mind $\omega_1$ being singular, then you can do just fine without assuming large cardinals.
The famous Feferman–Levy model is another example of this situation. In that model $\Bbb R$ is a countable union of countable sets. I think the original paper was only published as a notice in Notices of the AMS, but you can find modern presentations in Jech's books as well as numerous masters and doctoral theses over the year (e.g. Ioanna M. Dimitriou's theses both present the construction as an example of a symmetric extension).
Note that in any case, some use of forcing and symmetries is necessary. Depending on your set theoretic background, this might be a simple read (Solovay's paper is very readable), or an uphill battle.
| 169,113
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TITLE: Is long-term weather forecast impossible in principle?
QUESTION [8 upvotes]: This question can be asked about any chaotic dynamical system, but hydrodynamics of the atmosphere makes it more concrete. Arnold describes his 1966 result as follows:
I have calculated the curvature of this group [diffeomorphism group in hydrodynamics] and even used it to show that weather prediction is impossible for periods longer than two weeks. In a month you lose 3 digits in the prediction, just because of the curvature. This instability is not the Euler instability, it’s not describing a chaotic attractor of Euler equations – but it comes from the same line of ideas.
How final are the two weeks? One could imagine collecting data of greater precision and getting a meaningful longer term forecast. But there seems to be a theoretical limit to increasing precision, as with the diffraction limit to optical resolution. At a too fine enough precision, positions and momenta can not both be specified even theoretically, and the classical description breaks down. Quantum effects are usually negligible at classical scales, but does this apply to chaotic classical systems? In them initial discrepancies quickly magnify. Does this mean that quantum effects become classically relevant and long-term prediction of such systems is impossible in principle? Is there a theoretical time limit on weather forecasts for example?
Practical limits are discussed in How to calculate the upper limit on the number of days weather can be forecast reliably? Apparently, 15 days come up as well.
REPLY [3 votes]: I am not exactly sure what your question is (and which of sentences ending with a question mark are rethorical questions), so I try to answer them one at a time (and skip some questions that I consider to be answered by the answers to others):
Quantum effects are usually negligible at classical scales, but does this apply to chaotic classical systems?
There is no such thing as a purely classical system in reality, so quantum effects apply to all chaotic systems. Whether these effects are the predominant sources of inaccuracy depends on what exactly you regard as the system, more specifically whether there are any bigger influences (such as humans) that you do not consider part of the system.
On the other hand, in a theoretical purely classical system, there are no quantum effects by definition, but I suppose that’s not what you wanted to know.
Does this mean that quantum effects become classically relevant and long term prediction of such systems is impossible in principle?
Yes. Suppose, your system is the entire universe and you can measure everything as exactly as permitted by quantum effects. Further suppose that you are “outside” the universe, i.e., you are not part of the system yourself and your measurements and prediction efforts do not influence the system. Finally suppose that you have sufficent knowledge of physics to run a simulation as precise as permitted by your measurements. Then eventually quantum effects will affect macroscopic predictions by means of your simulation.
The reason for this is that the universe almost certainly has a positive Lyapunov exponent: It certainly contains a myriad of subsystems that have a positive Lyapunov exponent when isolated and there is no reason to assume that “coupling” them with the rest of the universe will change this in all cases.
Is there a theoretical time limit on weather forecasts for example?
Yes, though its exact value depends on what you consider to be part of your theory/system, e.g., are human behaviour or solar fluctuations part of your theory? Is your forecasting happening outside the system? Another factor is what exactly you consider a successful wheather forecast. If all those things are known, then there is an upper limit, but giving any reasonable value for it will probably be a huge (largely pointless) piece of work.
| 49,397
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Engineered – some would say OVER engineered – to withstand pretty much anything you could ever imagine throwing at a smart phone today, the Ulefone ARMOR 4G Smartphone is a flagship style Android device that is powerful, portable, and designed to take a licking and keep on ticking no matter what!
Sure, there are other flagship smart phones available on the market today that promise waterproofing, that promise shock-proofing, and that promise to withstand dents, dings, drops, and damage, but none of them (and we mean NONE of them) are as capable right out of the box as the Ulefone ARMOR 4G Smartphone is.
Initial impressions of Ulefone ARMOR 4G Smartphone
The Ulefone ARMOR 4G Smartphone is very much a smart phone designed for adventuring and those that themselves don’t mind getting a little bit down and dirty.
100% waterproof – and certified with IP 68 waterproofing standards – you can drop your phone in a river and leave it there for 30 minutes without anything going wrong with it. Not only that, but you can pull it out of that same river and take a phone call or send a text without even having to dry it off!
The same waterproofing features help to keep dust and debris out of this phone as well. This is a device that is as advanced as they come, providing you with the power and performance you’d expect from a flagship phone, but with a design intended for those that spend life in the fast lane.
Ulefone ARMOR Power and performance
The heartbeat of the Ulefone ARMOR 4G Smartphone is the brand-new eight core Snapdragon 1.3 GHz processor, the same kind of next-generation processor responsible for powering Android smart phones from some of the biggest companies on the planet.
Combine that with 3 GB of RAM and 32 GB of onboard storage (with the capacity for an extra 128 GB of storage with the use of a microSD card) and you are really rocking and rolling here!
The Ulefone ARMOR 4G Smartphone runs Android 6.0 right out of the box and will ALWAYS update to the latest version of Android the moment that it is released by Google. All of your favorite apps are just a few moments away from being downloaded, and thanks to the next generation Bluetooth, Wi-Fi, NFC, and 4G LTE connections you won’t ever have to worry about data transfer slowing down – regardless of where you use this phone around the world!
Ulefone ARMOR Standout features
As if all of that wasn’t enough to sway you into purchasing the Ulefone ARMOR 4G Smartphone – which, by the way, costs a fraction of what any other flagship Android phone will run you from the major manufacturers out there right now – you’re also going to be able to utilize a 13 megapixel camera, a 3500 mAh battery that lasts at least 18 hours on a single charge, and that’s just the tip of the iceberg.
The USB C connection guarantees lightning fast data transfer AND the ability to recharge your phone from completely dead to 75% powered up in about 45 minutes and then you’ll be able to get the rest of the way to 100% charged in about an hour after that.
It really doesn’t get much better than this!
Final verdict
At the end of the day, if you are serious about getting your hands on a flagship style Android smart phone without having to fork over $500 or more – and want the ability to utilize cutting-edge technology, 100% waterproofing, and a phone that can withstand everything short of a tank driving over it – the Ulefone ARMOR 4G Smartphone is the choice for you!
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TITLE: Why { $z-x-y=0$ , $z-2x=0$ , $2x+y-3z=0$ } cannot be solved this way?
QUESTION [1 upvotes]: I was recently solving a system of linear equations, 3 equations and 3 unknowns. I first solved via Row Reduction of the matrix and got a valid answer, but my friend attempted to solve the system using informal algebra methods and got the wrong answer. I know his answer is wrong, but I am struggling to explain what mathematical rule he broke.
Here is the system:
$z-x-y=0$
$z-2x=0$
$2x+y-3z=0$
Combining the first and third equation, one gets $x=-2y$. Plugging this back into equation one, one gets $z=-y$. Setting $x=1$, one gets the vector $<1,-1/2,1/2>$. This vector is valid for equations 1 and 3, but not for equation 2.
Now I know that this is not the proper technique for solving a system of three variables and that equation 2 was not used so how should one expect it to be satisfied. I know that this solution is wrong, but I am unsure how to explain what is wrong about it other than saying "that's not the way it's done." I personally made this mistake when first learning linear algebra and "that's not the way it's done" is all my teacher could say. If anyone has a better explanation for what exactly is wrong about this, I would greatly appreciate. Also, since equation 2 is not being used, it is 2 equations, 3 unknowns so there should be 2 free variables, not one (again I think this will occur if elimination is "done correctly").
REPLY [2 votes]: Let me present an extreme version of your friend's argument:
By the first equation, we have
$$ z = x + y $$
Then, simply choose $ x = y = 1 $ and conclude $ z = 2 $.
Thus, $(1,1,2)$ is a solution to the system!
Or to a ridiculous degree:
By ignoring all equations, we can choose $x = y = z = 1$ and so $(1,1,1)$ is a solution!
I hope this illuminates the issue with your friend's solution. Each equation potentially restricts the possible solutions to the system, and so each equation not used potentially un-restricts those solutions to ones which are not actually valid.
One more example to demonstrate this fact:
Consider the system of two equations for one variable,
$$ \begin{align*} x &= 1 \\ x &= 2 \end{align*} $$
By limiting our scope to only one equation, we may conclude either that $1$ is a solution or that $2$ is a solution, but in reality there are (obviously) no solutions, and so both are wrong.
| 190,349
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Pilgrimage Music Festival takes place September 22 -23 at Harlinsdale Farm in Franklin. If you don’t have tickets to see Chris Stapleton, Jack White, Lionel Richie, and more, here’s your chance to win free tickets.
City of Franklin is holding #MyPilgrimagePal contest now until September 7. It’s easy to enter – just post a photo of you and your pal with the hashtag #mypilgrimagepal on the City of Franklin Facebook page. Then write in 50 words or less why you should win.
The winner will be announced on September 12 on “Moore with the Mayor” live at 3:30 p.
Only legal residents of Tennessee are eligible to enter the contest. You can only enter the contest one time. See the complete rules and regulations here.
Watch the video below to learn about the Pilgrimage contest plus information on a music crawl happening the Friday night before Pilgrimage Festival (Sept 21). There will be over 20 musical acts performing in the Main Street area with venues like Kimbro’s and GRAYS on Main offering music into the evening. All of the acts performing entered a contest for the chance to perform at the new pre-Pilgrimage event.
Learn more about Franklin’s music festival – Pilgrimage Festival here.
| 245,801
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\begin{document}
\title{Falsification and future performance}
\author{
David Balduzzi\\
MPI for Intelligent Systems, T{\"u}bingen, Germany. \\
\texttt{david.balduzzi@tuebingen.mpg.de}
}
\maketitle
\begin{abstract}
We information-theoretically reformulate two measures of capacity from statistical learning theory: empirical VC-entropy and empirical Rademacher complexity. We show these capacity measures count the number of hypotheses about a dataset that a learning algorithm \emph{falsifies} when it finds the classifier in its repertoire minimizing empirical risk. It then follows from that the future performance of predictors on \emph{unseen} data is controlled in part by how many hypotheses the learner falsifies. As a corollary we show that empirical VC-entropy quantifies the message length of the true hypothesis in the optimal code of a particular probability distribution, the so-called actual repertoire.
\end{abstract}
\section{Introduction}
This note relates the number of hypotheses falsified by a learning algorithm to the expected future performance of the predictor it outputs. It does so by reformulating two basic results from statistical learning theory information-theoretically.
Suppose we wish to predict an unknown physical process $\sigma^*:\X\rightarrow \Y$ occurring in nature after observing its outputs $(y_1,\ldots, y_l)$ on sample $\data=(x_1,\ldots,x_l)$ of its inputs, where inputs arise according to unknown distribution $P$. One method is to take a repertoire $\cF$ of functions from $\X\rightarrow \Y$ and choose the predictor $\hat{f}\in\cF$ that best approximates $\sigma^*$ on the observed data. How confident can we be in $\hat{f}$'s future performance on unseen data?
Statistical learning theory provides bounds on $\hat{f}$'s expected future performance by quantifying a tradeoff implicit in the choice of repertoire $\cF$. At first glance, the bigger the repertoire the better since the best approximation to $\sigma^*$ in $\cF$ can only improve as more more functions are added to $\cF$. However, increasing $\cF$, and improving the approximation on observed data, can \emph{reduce} future performance due to overfitting. As a result, the bounds depend on both the accuracy with which $\hat{f}$ approximates $\sigma^*$ on the observed data and the capacity of repertoire $\cF$, see Theorems~\ref{t:vc} and \ref{t:rademacher}.
We wish to connect statistical learning theory with Popper's ideas about falsification. Popper argued that no amount of positive evidence confirms a theory \cite{popper:59}. Rather, theories should be judged on the basis of how many hypotheses they falsify. A theory is \emph{falsifiable} if there are possible hypotheses about the world (i.e. data) that are not consistent with the theory. A bold theory falsifies (disagrees with) many potential hypotheses about observed data. Testing a bold theory, by checking that the hypotheses it disagrees with are in fact false, provides corroborating evidence. If a theory has been thoroughly tested then (perhaps) we can have confidence in its predictions. Popper's criticism of positive confirmation was devastating. However, and hence the ``perhaps'', he failed to provide a rationale for trusting the predictions of severely tested theories.
To understand how falsifying hypotheses affects future performance we reformulate learning as a kind of \emph{measurement}. Before doing so, we need to describe precisely what we mean by measurement.
Given physical system $X$ with state space $S(X)$, a classical measurement is a function $f:S(X)\rightarrow \bR$. For example a thermometer $f$ maps configurations (positions and momenta) of particles in the atmosphere to real numbers. When the thermometer outputs $15^\circ C$ it generates information by specifying that atmospheric particles were in a configuration in $f^{-1}(15)\subset S(X)$. The information generated by the thermometer is a brute physical fact depending on how the thermometer is built and its output. We quantify the information, see \S\ref{s:meas}, by comparing the size of the total configuration space $S(X)$ with the size of the pre-image $f^{-1}(15)$. The smaller the pre-image, the more informative the measurement, see \S\ref{s:meas} for details.
More generally, any (classical) physical process $f:\X\rightarrow \Y$ can be thought of as performing measurements by taking inputs in $\X$ to outputs in $\Y$. Section~\S\ref{s:lim} introduces an important example, the \emph{min-risk} $\errmap_{\cF,\data}:\Sigma(\X,\Y)\rightarrow \bR$, which outputs the minimum value of the empirical risk over repertoire $\cF$ on a hypothesis space $\Sigma(\X,\Y)$. Finding the min-risk is a necessary step in finding the best approximation $\hat{f}$ to $\sigma^*$ in $\cF$. Since computing the min-risk requires actually implementing it as a physical process somehow or other, the measurements it performs and the effective information it generates are brute physical facts, no different in kind than the information generated by a thermometer.
It turns out that the min-risk categorizes hypotheses in $\Sigma$ according to how well they are approximated by predictors in repertoire $\cF$. Proposition~\ref{t:ei-vc} shows that the effective information generated by the min-risk is (essentially) the empirical VC-entropy. Moreover, the effective information generated by the min-risk ``counts'' the number of hypotheses about $\data$ that $\cF$ falsifies, see Eq.~\eqref{e:falsify}. As a consequence, Corollary~\ref{t:ei-vcb}, we obtain that the future performance of predictor $\hat{f}$ is controlled by {(\rm i)} how well $\hat{f}$ fits the observed data; {(\rm ii)} how many hypotheses about the data the min-risk rules out and {(\rm iii)} a confidence term.
It follows that, assuming the assumptions of the theorems below hold, bounds on future performance are brute physical facts resulting from the act of minimizing empirical risk, and so falsifying potential hypotheses, on observed data.
A consequence of our results, Corollary \ref{t:mml}, is that empirical VC-entropy is essentially the minimal length of the true hypothesis under the optimal code for the actual repertoire (a distribution depending on the min-risk). This suggests there may be interesting connections between VC-theory and the minimum message length (MML) approach to induction proposed by Wallace and Boulton \cite{wallace:68, wallace:05}.
Finally, section \S\ref{s:rrad} reformulates empirical Rademacher complexity via falsification. Here we build on Solomonoff's probability distribution introduced in \cite{solomonoff:64}. In short, we take Solomonoff's definition and substitute the \emph{min-risk} in place of the universal Turing machine, thereby obtaining what we refer to as the Rademacher distribution -- a \emph{non-universal} analog of Solomonoff's distribution. Rademacher complexity is then computed using the expectation of the min-risk over the Rademacher distribution, see Proposition \ref{t:mr-rad}.
The min-risk thus provides a bridge that not only connects VC-theory to a computable analog of Solomonoff's seminal distribution, but also sheds light on how falsification provides guarantees on future performance.
\textbf{Related work.}
The connection between Popper's ideas on falsifiability and statistical learning theory was pointed out in \cite{vapnik:98, corfield:09, harman:07}. However, these works focus on VC-dimension, which does not relate to falsification as directly as VC-entropy and Rademacher complexity which we consider here. Further, VC-entropy is a more fundamental concept in statistical learning theory than VC-dimension since VC-dimension is defined in terms of the limit behavior of the growth function, which is an upper bound on VC-entropy \cite{vapnik:98}. For more details on the link between MML and algorithmic probability, see \cite{wallace:99}.
\textbf{Acknowledgements.}
I thank David Dowe and Samory Kpotufe for useful comments on an earlier version of this paper.
\section{Measurement}
\label{s:meas}
We consider a toy universe containing probabilistic mechanisms (input/output devices) of the following form
\begin{defn}
Given finite sets $\X$ and $\Y$, a \textbf{mechanism} is a Markov matrix $\fm$ defined by conditional probability distribution $p_\fm(y|x)$.
\end{defn}
Mechanisms generate information about their inputs by assigning them to outputs \cite{bt:08, bt:09}.
\begin{defn}
The \textbf{actual repertoire} (or \textbf{measurement}) specified by $\fm$ outputting $y$ is the probability distribution
\begin{equation*}
p_\fm(x|y):=\frac{p_\fm(y|x)}{p(y)}\cdot p_{unif}(x),
\end{equation*}
where $p_{unif}(x)=\frac{1}{|\X|}$ is the uniform distribution. The \textbf{effective information} generated by the measurement is
\begin{equation*}
ei(\fm,y):=H\Big[p_\fm(X|y)\Big\|p_{unif}(X)\Big],
\end{equation*}
where $H[p\|q]=\sum_i p_i\log_2\frac{p_i}{q_i}$ is Kullback-Leibler divergence.
\end{defn}
The Kullback-Leibler divergence $H[p\|q]$ can be interpreted informally as the number of Y/N questions needed to get from distribution $q$ to distribution $p$. However, as pointed out in \cite{dowe:10}, Kullback-Leibler divergence is invariant with respect to the ``framing of the problem'' -- the ordering and structure of the questions -- suggesting it is a suitable measure of information-theoretic ``effort''.
The definition of measurement is motivated by the special case where $p_\fm$ assigns probabilities that are either 0 or 1; in other words, when it corresponds to a set-valued function $f:\X\rightarrow \Y$. The measurement performed by $f$ is
\begin{equation*}
p_f(x|y) = \left\{\begin{matrix}
\frac{1}{|f^{-1}(y)|} & \mbox{ if }f(x)=y\\
0 & \mbox{ else,}
\end{matrix}\right.
\end{equation*}
where $|\cdot|$ denotes cardinality. The support of $p_f(X|y)$ is the preimage $f^{-1}(y)\subset \X$. All elements of the support are assigned equal probability -- they are treated as an undifferentiated list. The measurement $p_\fm(X|y)$ therefore generalizes the notion of preimage to the probabilistic setting.
The effective information generated by $f$ outputting $y$ is $ei(f,y) = \log_2\frac{|\X|}{|f^{-1}(y)|}$:
\begin{equation}
\begin{matrix}
ei(f,y) &= & \log_2|\X|& -& \log_2|f^{-1}(y)|\\
&=& \Big(\mbox{no. potential inputs}\Big) & - & \Big(\mbox{no. inputs in pre-image}\Big) \\
&=& \Big(\mbox{no. inputs ruled out}\Big),
\end{matrix}
\label{e:det-ei}
\end{equation}
where inputs are counted in bits (after logarithming). Effective information is maximal ($\log_2|\X|$ bits) when a single input leads to $y$, and is minimal (0 bits) when \emph{all} inputs lead to $y$. In the first case, observing $f$ output $y$ tells us exactly what the input was, and in the latter case, it tells us nothing at all.
\begin{figure}[thpb]
\centering
\includegraphics[scale=.7]{measurement.pdf}
\caption{\textbf{The effective information generated by measurements.}
\footnotesize{
(A) A deterministic device can receive 144 inputs and produce 3 outputs. (B): Each input is implicitly assigned to a category (shaded areas). The information generated by the dark gray output is $\log_2 144-\log_2 9= 4$ bits.
}}
\label{f:meas}
\end{figure}
\subsection{Semantics}
\label{s:semantics}
Next we consider two approaches to characterizing the meaning of measurements. The first relates to possible world semantics \cite{lewis:86}. Here, the meaning of a sentence is given by the set of possible worlds in which it is true. Meaning is thus determined by considering all counterfactuals. For example, the meaning of ``That car is 10 years old'' is the set of possible worlds where the speaker is pointing to a car manufactured 10 years previously. Since the set of contains cars of many different colors, we see that color is irrelevant to the meaning of the sentence.
More precisely, the meaning of sentence $\cS$ is a map from possible worlds $W$ to truth values $v_\cS:W\rightarrow\{0,1\}$. Equivalently, the meaning of a sentence is
\begin{equation}
\begin{matrix}
W & \supset & v_\cS^{-1}(1)\\
\Big(\mbox{possible worlds}\Big) & \supset
& \Big(\mbox{worlds where }\cS\mbox{ is true}\Big).
\end{matrix}
\end{equation}
Inspired by possible world semantics, we propose
\begin{defn}
\label{d:mgs}
The \textbf{meaning} of output $y$ by mechanism $\fm$ is
\begin{equation}
\begin{matrix}
p_{unif}(X) & \rightarrow & p_\fm(X|y)\\
\Big(\mbox{possible inputs}\Big) & \rightarrow &
\Big(\mbox{inputs that cause }y\Big).
\end{matrix}
\end{equation}
For a deterministic function this reduces to $\X\supset f^{-1}(y)$.
\end{defn}
Grounding meanings in mechanisms yields four advantages over the possible worlds approach. First, it replaces the difficult to define notion of a possible world with the concrete set of inputs the mechanism is physically capable of receiving. Second, in possible world semantics the work of determining whether or not a sentence is true is performed somewhat mysteriously offstage, whereas the meaning of a measurement is determined via Bayes' rule. Third, the approach generalizes to probabilistic mechanisms. Finally, we can compute the effective information generated by a measurement, whereas there is no way to quantify the information content of a sentence in possible world semantics.
\subsection{Risk}
\label{s:pragmatics}
The second, pragmatic notion of meaning characterizes usefulness. We consider a special case, well studied in statistical learning theory, where usefulness relates to predictions \cite{vapnik:98}.
Let $\Sigma(\X,\Y)=\{\sigma:\X\rightarrow \Y\}$ be the set of all functions (deterministic mechanisms) mapping $\X$ to $\Y=\{-1,+1\}$. We will often write $\Sigma$ for short. Suppose there is a random variable $X$ taking values in $\X$ with unknown distribution $P$ and an unknown mechanism $\sigma^*\in\Sigma$, the \emph{supervisor}, who assigns labels to elements of $\X$.
\begin{defn}
The \textbf{risk} quantifies how well mechanism $f$ approximates an unknown or partially known mechanism $\sigma^*$:
\begin{equation}
\label{e:risk}
\risk(f) = \sum_{x\in \X} \bI\big[f(x)\neq \sigma^*(x)\big]\cdot p(x).
\end{equation}
It is the probability that $f$ and $\sigma^*$ disagree on elements of $\X$.
\end{defn}
Unfortunately, the risk cannot be computed since $P$ and $\sigma^*$ are unknown.
\begin{defn}
Given a finite sample $\data=(x_1,\ldots,x_l)\in\X^l$ with labels $\slabel=\sigma^*\data=(y_1,\ldots,y_l)\in\Y^l$, the \textbf{empirical risk} of $f:\X\rightarrow\Y$
\begin{equation}
\label{e:emp-risk}
\risk(f,\data,\slabel)=\frac{1}{l}\sum_{i=1}^l\bI\big[f(x_i)\neq y_i\big]
\end{equation}
is the fraction of the data $\data$ on which $f$ and $\sigma^*$ disagree.
\end{defn}
The empirical risk provides a computable approximation to the (true) risk.
\begin{rem}
\label{r:finite}
Note that in this paper, sets $\X$ and $\Y$ are both finite. Similarly, the training data $\data\in \X^l$ and labels $\slabel\in\Y^l$ also live in finite sets.
\end{rem}
\section{Statistical learning theory}
\label{s:slt}
Suppose we wish to predict the unknown supervisor $\sigma^*$ based on its behavior on labeled data $(\data,\slabel)$. A simple way to find a mechanism in repertoire $\cF\subset \Sigma(\X,\Y)$ that approximates $\sigma^*$ well is to minimize the empirical risk.
\begin{defn}
Given repertoire $\cF\subset\Sigma$ and unlabeled data $\data\in\X^l$, define \textbf{learning algorithm}
\begin{equation}
\label{e:algol}
\cA_{\cF, \data}:\Sigma \rightarrow \cF:\sigma \mapsto \arg\min_{f\in\cF}\risk(f,\data,\sigma\data)
\end{equation}
which finds the mechanism in $\cF$ that minimizes empirical risk.
\end{defn}
Learning algorithm $\cA_{\cF,\data}$ finds the mechanism in $\cF$ that appears, based on the empirical risk, to best approximate $\sigma^*$. Empirical risk stays constant or decreases as $\cF$ is enlarged, suggesting that the larger the repertoire the better.
This is not true in general since minimizing risk -- and \emph{not} empirical risk -- is the goal. There is a tradeoff: increasing the size of $\cF$ leads to overfitting the data which can increase risk even as empirical risk is reduced.
The tendency of a repertoire to overfit data depends on its size or capacity. We recall two measures of capacity that are used to bound risk: empirical VC-entropy \cite{vapnik:82} and empirical Rademacher complexity \cite{koltchinskii:01}.
\begin{defn}
Given unlabeled data $\data\in\X^l$ and repertoire $\cF\subset\Sigma$ let
\begin{equation}
q_\data:\cF\rightarrow \bR^l:f\mapsto\Big(f(x_1),\ldots,f(x_l)\Big).
\label{e:vc-ent}
\end{equation}
The empirical \textbf{VC-entropy}\footnote{VC-entropy is the \emph{expectation} of empirical VC-entropy \cite{vapnik:98}. Also, note the standard definition of VC-entropy uses $\log_e$ rather than $\log_2$.} of $\cF$ on $\data$ is $\cV(\cF,\data):=\log_2 |q_\data(\cF)|$, where $|q_\data(\cF)|$ is the number of distinct points in the image of $q_\data$.
The empirical \textbf{Rademacher complexity} of $\cF$ on $\data$ is
\begin{equation}
\radem(\cF,\data) = \frac{1}{|\Sigma|}
\sum_{\sigma\in\Sigma}\left[\sup_{f\in\cF}\frac{1}{l}\sum_{i=1}^l\sigma(x_i)\cdot f(x_i)\right].
\label{e:rademacher}
\end{equation}
\end{defn}
VC-entropy ``counts'' how many labelings of $\data$ the classifiers in $\cF$ fit perfectly. Rademacher complexity is a weighted count of how many labelings of $\data$ functions in $\cF$ fit well.
The following theorems are shown in \cite{boucheron:00} and \cite{bousquet:04} respectively:
\begin{thm}[empirical VC-entropy bound]\label{t:vc}\eod
With probability $1-\delta$, the expected risk is bounded by
\begin{equation}
\risk(f) \leq \risk(f,\data,\slabel)
+ c_1\sqrt{\frac{\cV(\cF,\data)}{l}}
+ c_2\sqrt{\frac{1-\log_2\delta}{l}}
\label{e:bd-vc}
\end{equation}
for all $f\in \cF$, where the constants are $c_1=\sqrt{\frac{6}{\log_2 e}}$ and $c_2=\sqrt{\frac{1}{\log_2 e}}$.
\end{thm}
\begin{thm}[empirical Rademacher bound]\label{t:rademacher}\eod
For all $\delta>0$, with probability at least $1-\delta$,
\begin{equation}
\risk(f)\leq \risk(f,\data,\slabel)+ \radem(\cF,\data)+c_3\sqrt{\frac{1-\log_2\delta}{l}},
\label{e:bd-rad}
\end{equation}
for all $f\in \cF$, where $c_3=\sqrt{\frac{2}{\log_2 e}}$.
\end{thm}
The tradeoff between empirical risk and capacity is visible in the first two terms on the right-hand sides of the bounds.
The left-hand sides of Eqs~\eqref{e:bd-vc} and \eqref{e:bd-rad} cannot be computed since $P$ and $\sigma^*$ are unknown. Remarkably, the right-hand sides depend only on mechanism $f$ chosen from repertoire $\cF$, labeled data $(\data,\slabel)$ and desired confidence $\delta$. The theorems assume data is drawn \emph{i.i.d.} according to $P$ and labeled according to $\sigma^*$; it make no assumptions about the distribution $P$ on $\X$ or supervisor $\sigma^*$, except that they are \emph{fixed}.
\section{Falsification}
\label{s:lim}
This section reformulates the results from statistical learning theory to show how the past falsifications performed by a learning algorithm control future performance. We show that the empirical VC-entropies and Rademacher complexities admit interpretations as ``counting'' (in senses made precise below) the number of hypotheses falsified by a particular measurement performed when learning.
We start by introducing a special mechanism, the min-risk, which is used implicitly in learning algorithm $\cA_{\cF,\data}$. As we will see, the structure of the measurements performed by the min-risk determine the capacity of the learning algorithm.
\begin{defn}
\label{d:error}
Given repertoire $\cF\subset\Sigma$ and unlabeled data $\data\in\X^l$, define the \textbf{min-risk} as the minimum of the empirical risk on $\cF$:
\begin{equation}
\label{e:error}
\errmap_{\cF,\data}:\Sigma \rightarrow \bR:\sigma \mapsto\min_{f\in\cF} \risk(f,\data,\sigma\data).
\end{equation}
\end{defn}
The min-risk is a mechanism mapping supervisors $\sigma$ in $\Sigma$ to the empirical risk of their best approximations $\cA_{\cF,\data}(\sigma)$ in $\cF$, see Fig. \ref{f:mrisk}. Note that inputs to the min-risk are themselves mechanisms.
We suggestively interpret the setup as follows. Suppose a scientist studies a universe where inputs in $\X$ appear according to distribution $P$, and are assigned labels in $\Y$ by unknown physical process $\sigma^*$. The \emph{hypothesis space} is $\Sigma(\X,\Y)$, the set of all possible (deterministic) physical processes that take $\X$ to $\Y$.
The scientist's goal is to learn to predict physical process $\sigma^*$, on the basis of a small sample of labeled data $(\data,\slabel)$. She has a \emph{theory}, repertoire $\cF$, and a method, $\cA_{\cF,\data}$, which she uses to fit some particular $\hat{f}\in\cF$ given $\slabel$.
The most important question for the scientist is: How reliable are predictions made by $\hat{f}$ on \emph{new} data? We will show that $\hat{f}$'s reliability depends on the measurements performed by the min-risk -- i.e. on the work done by the scientist when she applies method $\cA_{\cF,\data}$ to find $\hat{f}$.
\begin{figure}[thpb]
\centering
\includegraphics[scale=.8]{min-risk.pdf}
\caption{\textbf{The structure of the measurement performed by the min-risk.}
\footnotesize{
The min-risk categorizes potential hypothesis in $\Sigma$ according to how well they are fit by mechanisms in theory $\cF$.
}}
\label{f:mrisk}
\end{figure}
\subsection{Empirical VC entropy}
\label{s:rvc}
Empirical VC-entropy is, essentially, the effective information generated by the min-risk when it outputs a perfect fit:
\begin{prop}[VC-entropy via effective information]\label{t:ei-vc}\eod
Empirical VC entropy is
\begin{equation}
\cV(\cF,\data) = l - ei\left(\errmap_{\cF,\data}, 0\right).
\label{e:ei-vc}
\end{equation}
\end{prop}
\noindent
Proof:
Let $\X=\data\cup\data^c$ and $|\X|=m$. Then $\Sigma=\{\sigma:\data\rightarrow \Y\}\times \{\sigma:\data^c\rightarrow \Y\}$. By definition
\begin{equation*}
ei\left(\errmap_{\cF,\data}, 0\right) = \log_2 |\Sigma| - \log_2| \errmap^{-1}_{\cF,\data}(0)|,
\end{equation*}
with $\log_2 |\Sigma|=m$. It remains to show that $| \errmap^{-1}_{\cF,\data}(0)|=2^{m-l}\cdot |q_\data(\cF)|$. Points in the image of $q_\data$ correspond to labelings $\sigma$ of the data by functions in $\cF$. Thus, $|q_\data(\cF)|$ counts distinct labelings of $\data$ that $\cF$ fits perfectly. These occur with multiplicity $2^{m-l}$ in the pre-image by the product decomposition of $\Sigma$ above.
$\blacksquare$
We interpret the result as follows. Suppose the scientist applies theory $\cF$ to explain her labeled data and perfectly fits function $\hat{f}=\cA_{\cF,\data}(\sigma^*)$ with risk $\epsilon=0$.
By Definition~\ref{d:mgs}, the meaning of her work is $\Sigma \supset \errmap_{\cF,\data}^{-1}(0)$: the set of mechanisms that her theory $\cF$ fits perfectly. The effective information generated by her work is
\begin{equation}
\begin{matrix}
ei(\errmap_{\cF,\data},0) & = & \log_2\left|\Sigma\right| & - & \log_2|\errmap_{\cF,\data}^{-1}(0)|\\
& = & \Big(\mbox{total no. of hypotheses}\Big) & - & \Big(\mbox{no. that theory fits}\Big) \\
& = & \Big(\mbox{no. of hypotheses falsified}\Big),
\end{matrix}
\label{e:falsify}
\end{equation}
where hypotheses are counted in bits (after logarithming). A theory is informative if it rules out many potential hypotheses \cite{popper:59}.
\emph{The number of hypotheses the scientist falsifies when using theory $\cF$ to fit $\hat{f}$ has implications for its future performance}:
\begin{cor}[information-theoretic empirical VC bound]\label{t:ei-vcb}\eod
With probability $1-\delta$, the risk of predictor $\hat{f}=\cA_{\cF,\data}(\sigma^*)$ outputted by learning algorithm $\cA_\cF$ is bounded by
\begin{equation}
\risk(f) \leq \risk(f,\data,\slabel) + c_1\sqrt{1-\frac{ei(\errmap_{\cF,\data},0)}{l}}
+ c_2\sqrt{\frac{1-\log_2 \delta}{l}}.
\label{e:ei-vcb}
\end{equation}
\end{cor}
\noindent
Proof: By Theorem~\ref{t:vc} and Proposition~\ref{t:ei-vc}.
$\blacksquare$
The corollary states that minimizing empirical risk embeds expectations about the future into predictors. So long as the corollary's assumptions hold, future performance by $\hat{f}$ is controlled by: {(\rm i)} the output of the min-risk, i.e. the fraction $\epsilon$ of the data that $\hat{f}$ fits; {(\rm ii)} the effective information generated by the min-risk, i.e. the number (in bits) of hypotheses the learning algorithm falsifies if it fits perfectly; and {(\rm iii)} a confidence term. The only assumption made by the corollary is that $P$ and $\sigma^*$ are \emph{fixed}.
\begin{rem}
The theorem provides no guarantees on the future performance of a theory that ``explains everything'', i.e. $\cF=\Sigma$, no matter how well it fits the data. This follows since effective information is zero when $\cF=\Sigma$, and so the second term on the right-hand side of Eq.~\eqref{e:ei-vcb} is $c_1\approx 2$.
\end{rem}
Reformulating the above result in terms of code lengths suggests a connection between VC-theory and minimum message length (MML), see \cite{wallace:68} and \S6.6 of \cite{dowe:10}. Recall that, given probability distribution $p(X)$, the message length of event $x$ in an optimal binary code is $\text{len}(x):=-\log_2 p(x)$.
\begin{cor}[VC-entropy controls code length of true hypothesis]\label{t:mml}\eod
Denote the min-risk by $\fm=\risk_{\cF,\data}$. The length of the true hypothesis $\hat{\sigma}$ in the optimal code for the actual repertoire specified by the min-risk, $p_\fm(\Sigma|\epsilon=0)$, is
\begin{equation*}
\text{len}(\hat{\sigma}) = \cV(\cF,\data) + \big(|\X|-|\data|\big).
\end{equation*}
\end{cor}
\noindent
Proof: By Proposition \ref{t:ei-vc} we have $-\log_2 p_\fm(\hat{\sigma}|\epsilon=0) = \log_2 |\risk_{\cF,\data}^{-1}(0)|$.
$\blacksquare$
The length of the message describing the true hypothesis in the actual repertoire's optimal code is the empirical VC-entropy plus a term, $(|\X|-|\data|)=(m-l)$, that decreases as the amount of training data increases. The shorter the message, the better the predictor's expected performance (for fixed empirical risk).
\subsection{Empirical Rademacher complexity}
\label{s:rrad}
VC-entropy only considers hypotheses that theory $\cF$ fits perfectly. Rademacher complexity is an alternate capacity measure that considers the distribution of risk across the entire hypothesis space. This section explains Rademacher complexity via an analogy with Solomonoff probability \cite{solomonoff:64, wallace:99}.
We first recall Solomonoff's definition. Given universal Turing machine $T$, define (unnormalized) \textbf{Solomonoff probability}
\begin{equation}
\label{e:solomonoff}
p_T(s) := \sum_{\{i|T(i)= s\bullet\}} 2^{-\text{len}(i)},
\end{equation}
where the sum is over strings\footnote{A technical point is that no proper prefix of $i$ should output $s$.} $i$ that cause $T$ to output $s$ as a prefix, and $\text{len}(i)$ is the length of $i$. We adapt Eq.~\eqref{e:solomonoff} by replacing Turing machine $T$ with min-risk $\risk_{\cF,\data}:\Sigma\rightarrow \bR$.
\begin{defn}
Equipping hypothesis space with the uniform distribution $p_{unif}(\Sigma)$, all hypotheses have length $\text{len}(\sigma)=|\X|=\log_2|\Sigma|$ in the optimal code. Set the \textbf{Rademacher distribution} for the min-risk $\fm=\risk_{\cF,\data}$ as
\begin{equation}
\label{e:rad_dist}
p_\fm(\epsilon) :=
\sum_{\left\{\sigma|R_{\cF,\data}(\sigma)=\epsilon\right\}}2^{-\text{len}(\sigma)}=\left\{\begin{matrix}
\frac{\big|\errmap_{\cF,\data}^{-1}(\epsilon)\big|}{|\Sigma|} &
\mbox{ if }\epsilon\in \errmap_{\cF,\data}(\Sigma)\\ \\
0 & \mbox{ else.}
\end{matrix}\right.
\end{equation}
\end{defn}
The Rademacher distribution is constructed following Solomonoff's approach after substituting the min-risk as a ``special-purpose Turing machine'' that only accepts hypotheses in finite set $\Sigma$ as inputs. It tracks the fraction of hypotheses in $\Sigma$ that yield risk $\epsilon$.
The Rademacher distribution arises naturally as the denominator when using Bayes' rule to compute the actual repertoire $p_\fm(\Sigma|\epsilon)$:
\begin{equation*}
p_\fm(\sigma|\epsilon)=\frac{p_\fm(\epsilon|\sigma)}{p_\fm(\epsilon)}\cdot p_{unif}(\sigma),
\,\,\,\text{ where }p_\fm(\epsilon|\sigma)=\left\{\begin{matrix}
1 &
\mbox{ if }\errmap_{\cF,\data}(\sigma)=\epsilon\\ \\
0 & \mbox{ else.}
\end{matrix}\right.
\end{equation*}
\begin{prop}[Rademacher complexity via min-risk]\label{t:mr-rad}\eod
\begin{equation}
\radem(\cF,\data)=1-2\cdot \bE\big[\epsilon\,\big|\,p_\fm(\epsilon)\big].
\end{equation}
\end{prop}
\noindent
Proof: We refer to $\bE\big[\epsilon\,\big|\,p_\fm(\epsilon)\big]$ as the expected min-risk. From Eq.~\eqref{e:rademacher},
\begin{equation*}
\radem(\cF,\data) = \frac{1}{|\Sigma|}
\sum_{\sigma\in\Sigma}
\left[\sup_{f\in\cF}\frac{1}{l}\sum_{i=1}^l\sigma(x_i)\cdot f(x_i)\right].
\end{equation*}
Observe that $\frac{1}{l}\sum_{i=1}^l\sigma(x_i)\cdot f(x_i) = 1-2\risk(f,\data,\sigma)$. It follows that $\sup_{f\in\cF}\frac{1}{l}\sum_{i=1}^l\sigma(x_i)\cdot f(x_i)=1-2\errmap_{\cF,\data}(\sigma)$, which implies
\begin{equation*}
\radem(\cF,\data) = 1-2\sum_{\sigma\in \Sigma}
\frac{\errmap_{\cF,\data}(\sigma)}{|\Sigma|}
= 1-2\sum_{\epsilon} \epsilon\cdot \frac{\big|\errmap_{\cF,\data}^{-1}(\epsilon)\big|}{|\Sigma|}.
\,\,\blacksquare
\label{e:rad-int}
\end{equation*}
\vspace{1mm}
Rademacher complexity is low if the expected min-risk is high. The expected min-risk admits an interesting interpretation. For any hypothesis $\sigma\in\errmap_{\cF,\data}^{-1}(\epsilon)$ the classifier $\hat{f}_\sigma:=\cA_{\cF,\data}(\sigma)\in\cF$ outputted by the learning algorithm yields incorrect answers on fraction $\epsilon=\frac{1}{l}\sum_{i=1}^l\bI\big[\hat{f}_\sigma(x_i)\neq \sigma(x_i)\big]$ of the data. It follows that
\begin{equation*}
\begin{matrix}
\sum_\epsilon p_\fm(\epsilon)\cdot \epsilon & = & \sum_\epsilon &
\frac{\big|\errmap_{\cF,\data}^{-1}(\epsilon)\big|}{|\Sigma|}
&\cdot&
\frac{1}{l}\sum_l \bI\big[\hat{f}_\sigma(x_i)\neq\sigma(x_i)\big]
\\
& = & \sum_\epsilon &\Big(\mbox{fraction of hypotheses falsified}\Big)&\cdot &\Big(\mbox{on fraction }\epsilon\mbox{ of the data}\Big).
\end{matrix}
\end{equation*}
A bold theory $\cF$ is one for which $\bE[\epsilon|p_\fm(\epsilon)]$ is high, meaning that its predictors (the classifiers it tries to fit to data) are sufficiently narrow that it would falsify most hypotheses on most of the data.
\emph{When a bold theory happens to fit labeled data well, it is guaranteed to perform well in future:}
\begin{cor}[information-theoretic empirical Rademacher bound]\label{t:ei-radb}\eod
With probability $1-\delta$, the risk of predictor $\hat{f}=\cA_\cF(\data,\slabel)$ outputted by learning machine $\cA_\cF$ is bounded by
\begin{equation}
\risk(f) \leq \risk(f,\data,\slabel)
+ \left[1-2\sum_{\epsilon} \epsilon\cdot 2^{-ei(\errmap_{\cF,\data},\epsilon)}\right]
+ c_3\sqrt{\frac{1-\log_2\delta}{l}}
\end{equation}
\end{cor}
Proof: By Proposition~\ref{t:mr-rad} and definition of effective information we have
\begin{equation*}
\radem(\cF,\data)=1-2\sum_{\epsilon} \epsilon\cdot \frac{\big|\errmap_{\cF,\data}^{-1}(\epsilon)\big|}{|\Sigma|}=1-2\sum_{\epsilon} \frac{\epsilon}{2^{ei(\errmap_{\cF,\data},\epsilon)}}.
\end{equation*}
The result follows by Theorem~\ref{t:rademacher}.
$\blacksquare$
Rademacher complexity is low if the min-risk's sharp measurements (high $ei$) are accurate (low $\epsilon$), and conversely. Analogously to Corollary~\ref{t:ei-vcb}, the Rademacher bound implies the future performance of a classifier depends on: {(\rm i}) the fraction $\epsilon$ of the data that $\hat{f}$ fits; {(\rm ii)} the weighted (by the fraction $\epsilon$ of data that falsifies them) sum of the fraction of hypotheses falsified; and {(\rm iii}) a confidence term. Once again, the only assumption is that $P$ and $\sigma^*$ are \emph{fixed}.
\section{Discussion}
\label{s:discuss}
Learning according to algorithm $\cA_{\cF,\data}$ entails computing the min-risk, which classifies hypotheses about $\data$ according to how well they are approximated by predictors in repertoire $\cF$. Repertoires that rule out many hypotheses when they fit labeled data $(\data,\slabel)$ generate more effective information than repertoires that ``approximate everything''. As a consequence, when and if an informative repertoire fits labeled data well, Corollary~\ref{t:ei-vcb} implies we can be confident in future predictions on unseen data.
A pleasing consequence of reformulating empirical VC-entropy and empirical Rademacher complexity in terms of falsifying hypotheses is that it directly connects Popper's intuition about falsifiable theories to statistical learning theory, thereby providing a rigorous justification for the former.
Our motivation for reformulating learning theory information-theoretically arises from a desire to better understand the role of information in biology. Although Shannon information has been heavily and successfully applied to biological questions, it has been argued that it does not fully capture what biologists mean by information since it is not semantic. For example, Maynard Smith states that ``In biology, the statement that A carries information about B implies that A has the form it does because it carries that information'' \cite{maynardsmith:00}. Shannon information was invented to study communication across prespecified channels, and lacks any semantic content. Maynard Smith therefore argues that a different notion of information is needed to understand in what sense evolution and development embed information into an organism.
It may be fruitful to apply statistical learning theory to models of development. One possible approach is to consider analogs of repertoire $\cF$. For example, $\cF$ may correspond to the repertoire of possible adult forms a zygote could develop into. The particular adult form chosen, $\hat{f}\in\cF$, depends on the historical interactions $(\data,\slabel)$ between the organism and its environment, assuming these can be suitably formalized. The information generated by the organism's development would then have implications for its future interactions with its environment. More speculatively, a similar tactic could be applied to quantify the information embedded in populations by inheritance and natural selection.
{
\footnotesize\small
\bibliographystyle{splncs03}
| 184,483
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\begin{document}
\title
{The first obstructions to enhancing a triangulated category}
\author{Fernando Muro}
\address{Universidad de Sevilla,
Facultad de Matemáticas,
Departamento de Álgebra,
Avda. Reina Mercedes s/n,
41012 Sevilla, Spain}
\email{fmuro@us.es}
\urladdr{http://personal.us.es/fmuro}
\thanks{The author was partially supported by the Spanish Ministry of Economy under the grant MTM2016-76453-C2-1-P (AEI/FEDER, UE)}
\subjclass
{}
\keywords{}
\begin{abstract}
In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an example of triangulated category without models. We also obtain cohomological characterizations of pre-triangulated DG, $A$-infinity, and spectral categories.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Heller \cite{heller_stable_1968} noted that a triangulated structure on an essentially small additive category $\C T$ with suspension functor $\Sigma\colon\C T\rightarrow\C T$ induces a stable \emph{Toda bracket} partial composition-like operation, sending morphisms
\begin{equation}\label{three_maps}
X\stackrel{f}{\To}Y\stackrel{g}{\To}Z\stackrel{h}{\To}T
\end{equation}
with $gf=0$ and $hg=0$ to a coset
\[\langle h,g,f\rangle\subset\C T(X,\Sigma^{-1}T),\]
satisfying certain properties. Exact triangles
\[X\stackrel{f}{\To}Y\stackrel{i}{\To}C_f\stackrel{q}{\To}\Sigma X\]
are characterized by the fact that the Toda bracket contains the identity map
\[\operatorname{id}_X\in \langle q,i,f\rangle\subset\C T(X,X).\]
The graded category $\C T_{\Sigma}$ associated with the pair $(\C T, \Sigma)$ is given by
\[\C T_{\Sigma}^n(X,Y)=\C T(X,\Sigma^nY),\quad n\in\mathbb Z.\]
An \emph{algebraic enhancement} of $\C T$ in the sense of Bondal and Kapranov \cite{bondal_enhanced_1991} is a DG-category $\C C$ with $H^*(\C C)=\C T_{\Sigma}$ such that the previous Toda brackets coincide with the standard Massey products in the cohomology of $\C C$.
Assume we are working over a field $k$. By Kadeishvili's theorem \cite{kadeishvili_theory_1980, lefevre-hasegawa_sur_2003}, a Bondal--Kapranov enhancement is the same as a minimal $A$-infinity category structure on $\C T_{\Sigma}$. The first possibly non-trivial piece of this structure is a multilinear ternary composition operation $m_3$, defined on chains of three composable morphisms like \eqref{three_maps} without further conditions, such that
\[m_3(h,g,f)\in\C T(X,\Sigma^{-1}T)\]
is a well-defined element. The connection with the triangulated structure is that
\[m_3(h,g,f)\in\langle h,g,f\rangle\]
whenever the Toda bracket is defined. The compatibility properties between $m_3$ and composition in $\C T$ amount to saying that $m_3$ is a Hochschild cocycle. We denote its cohomology class by
\[\{m_3\}\in \hh{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\]
and call it \emph{universal Massey product}. This class, previously considered in e.g.~\cite{kadeishvili_algebraic_1982, benson_realizability_2004}, is independent of the choice of minimal model.
Two natural questions arise: Does a given triangulated category have an enhancement? If so, how many essentially different ones? There are remarkable results on the existence and uniqueness of enhancements for certain triangulated categories \cite{lunts_uniqueness_2010, canonaco_uniqueness_2015} as well as examples which do not admit any enhancement \cite{muro_triangulated_2007, dimitrova_triangulated_2009, rizzardo_k-linear_2018}. There are even examples with essentially different algebraic enhancements over a field \cite{kajiura_-enhancements_2013} and some others where the first question remains open \cite{amiot_structure_2007}. In this paper we give the first step towards a different, obstruction-theoretic approach applicable to any $\C T$.
We start by considering the set of triangulated structures on a pair $(\C T,\Sigma)$ as above. For the moment we only consider triangulated structures in the sense of Puppe \cite{puppe_formal_1962}, i.e.~not requiring Verdier's octahedral axiom. Freyd \cite{freyd_stable_1966} proved that, for that set to be non-empty, the category $\modulesfp{\C T}$ of finitely presented right $\C T$-modules must be a Frobenius abelian category. In that case we can define the stable module category $\modulesst{\C T}$, which is canonically triangulated with suspension functor $S$, the cosyzygy functor, and $\Sigma$ induces a triangulated endofunctor of $\modulesst{\C T}$. If in addition idempotents split in $\C T$, Heller \cite{heller_stable_1968} defined a bijection between the set of triangulated structures on $(\C T,\Sigma)$ and a subset of the set of natural transformations $\Sigma\rightarrow S^3$ between endofunctors of $\modulesst{\C T}$ satisfying two algebraic conditions.
The requirement on idempotents is harmless because any triangulated structure extends uniquely to the idempotent completion of $\C T$ \cite{balmer_idempotent_2001}.
In Section \ref{heller_section} we identify the set of triangulated structures with a subset of the Hochschild cohomology group
\[\hh{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}},\]
where $\modulesfp{\C T_{\Sigma}}$ is the graded abelian category of finitely presented right $\C T_{\Sigma}$-modules. This group is usually strictly smaller than Heller's set of natural transformations $\Sigma\rightarrow S^3$. The subset of triangulated structures is again defined by two algebraic conditions on the Hochschild cohomology classes which correspond to Heller's. The second condition is identified later, in Section \ref{spectral_sequence_section} for triangulated categories over a field, and in Section \ref{topological_section} in general.
We prove in Section \ref{toda_brackets_section} that the previous Hochschild cohomology group is in bijection with the set of all stable Toda brackets in $(\C T,\Sigma)$. We also show that unstable Toda brackets are in bijection with another (ungraded) Hochschild cohomology group where the former injects. The latter is also in bijection with the set of natural transformations $\Sigma\rightarrow S^3$ used by Heller. This establishes a direct link between Toda brackets and natural transformations $\Sigma\rightarrow S^3$ not considered by Heller in \cite{heller_stable_1968} despite he used both.
In Sections \ref{spectral_sequence_section} and \ref{octahedral_axiom_section} we restrict ourselves to working over a ground field $k$. In the first one we define a first quadrant spectral sequence of graded vector spaces
\[E_2^{p,q}=\hh{p,\ast}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{q,\ast}}\Longrightarrow\hh{p+q,\ast}{\C T_{\Sigma}, \C T_{\Sigma}}\]
whose edge morphism
\begin{equation}\label{edge}
\hh{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\To \hh{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}
\end{equation}
takes the universal Massey product of any enhancement to the stable Toda bracket of the triangulated structure,
\[\{m_3\}\mapsto \langle-,-,-\rangle.\]
The spectral sequence is not totally new, a related ungraded version has been considered in \cite{lowen_hochschild_2005}.
Using the connection between the stable Toda bracket of a triangulated category and the universal Massey product of any possible enhancement via the previous spectral sequence, we can define the first obstructions to the existence of a Bondal--Kapranov enhancement. The very first obstruction is the image of the Toda bracket along the spectral sequence differential
\begin{equation}\label{d2}
d_2\colon \hh{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}\To \hh{2,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{2,\ast}}.
\end{equation}
If it vanishes, then $\langle-,-,-\rangle\in E_3^{0,3}\subset \hh{0,\ast}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}$ is in the third page of the spectral sequence and the second obstruction is its image along
\[d_3\colon E_3^{0,3}\To E^{3,1}_3.\]
The second obstruction vanishes when $\langle-,-,-\rangle\in E_4^{0,3}\subset E_3^{0,3}$ is in the fourth page. In this case there is a third obstruction, the image of the Toda bracket under
\[d_4\colon E_4^{0,3}\To E^{4,0}_4.\]
When it vanishes, $\langle-,-,-\rangle\in E_\infty^{0,3}=E_5^{0,3}\subset E_4^{0,3}$ is a permament cycle, i.e.~the Toda bracket is in the image of the edge morphism \eqref{edge}. If this happens, any preimage is a potential universal Massey product, that is to say, if a representing cocycle $m_3$ can be extended to a full minimal $A$-infinity algebra structure on $\C T_{\Sigma}$, then this extension is an enhancement for the triangulated structure on $\C T$. Abusing terminology, we will sometimes say that a triangulated category over a field has a universal Massey product if its stable Toda bracket has a preimage along the edge morphism \eqref{edge}.
There is a well-known classical obstruction theory with values in Hochschild cohomology for the extension of truncated minimal $A$-infinity category structures, see e.g.~\cite{lefevre-hasegawa_sur_2003}. We have developed an enhanced version in \cite{muro_enhanced_2015}, after Angeltveit \cite{angeltveit_topological_2008}. It will be further investigated in the triangulated context in a subsequent paper.
Any Puppe triangulated category with an enhacement satisfies Verdier's octahedral axiom. However, it does not seem to be possible to translate this axiom into algebraic conditions on Hochschild cohomology classes. In Section \ref{octahedral_axiom_section} we give a sufficient algebraic condition for the octahedral axiom: it holds provided the Toda bracket classifying the triangulated structure is in the kernel of the spectral sequence differential \eqref{d2}. In particular, any triangulated structure over a field with a universal Massey product satisfies the octahedral axiom. This also leads to an apparently new homological characterization of pre-triangulated DG- and $A$-infinity categories over a field in the sense of \cite{bondal_enhanced_1991}.
Many triangulated categories do not have algebraic enhancements. The most prominent example is the stable homotopy category of spectra. Nevertheless, most triangulated categories have a \emph{topological enhancement}, which consists of a full embedding into the homotopy category $\operatorname{Ho}\C M$ of a stable model category $\C M$ \cite{schwede_algebraic_2010,hovey_model_1999}. In the topological context, enhancements are usually called \emph{models}. Algebraic enhancements are also topological.
Using \cite{baues_homotopy_2007}, we show in Section \ref{topological_section} that a topological enhancement of $\C T$ gives rise to a cohomology class
\[\{m_3\}\in \hbw{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\]
in the non-additive cohomology of the category $\C T_{\Sigma}$, which is a generalization of Mac Lane and topological Hochschild cohomology of rings. This cohomology class is called \emph{universal Toda bracket}. Formally, a cocycle $m_3$ representing the universal Toda bracket is a ternary operation as above, except for the fact that it need not be multiliear. The connection to stable Toda brackets in the triangulated category $\C T$ is exactly as above. Universal Toda brackets were introduced in \cite{baues_cohomology_1989}, where their connection with ordinary Toda brackets is also established.
Hochschild cohomology coincides with its non-additive counterpart in dimension $0$, in particular
\[\hh{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}\cong \hbw{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}},\]
so both of them are in bijection with stable Toda brackets in $(\C T,\Sigma)$.
In Section \ref{topological_section} we define a first quadrant spectral sequence of graded abelian groups for non-additive cohomology of categories
\[E_2^{p,q}=\hbw{p,\ast}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{q,\ast}}\Longrightarrow\hbw{p+q,\ast}{\C T_{\Sigma}, \C T_{\Sigma}}\]
whose edge morphism
\begin{equation}\label{edge_top}
\hbw{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\To \hbw{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}
\end{equation}
takes the universal Toda bracket of any topological enhancement to the stable Toda bracket of the triangulated structure,
\[\{m_3\}\mapsto \langle-,-,-\rangle.\]
An ungraded version of this spectral sequence appears in \cite{jibladze_cohomology_1991} with an apparently different target.
We can define the first three obstructions to the existence of a topological enhancement as above. We also show that the previous sufficient condition for Verdier's octahedral axiom is still valid for this spectral sequence. In particular any triangulated category with a universal Toda bracket satisfies the octahedral axiom. This leads to an new homological characterization of triangulated spectral categories in the sense of Tabuada \cite{tabuada_matrix_2010}. Moreover, it shows that, if we ever find a Puppe triangulated category which does not satisfy the octahedral axiom, then the very first obstruction to the existence of a topological enhancement must be non-zero.
We conclude in Section \ref{example} with an explicit example where the obstructions do not vanish. The triangulated category $\C T$ in the example is the category of finitely generated free modules over $\mathbb Z/4$ with $\Sigma$ the identity functor. This category is among the first known examples of triangulated categories without models of any kind \cite{muro_triangulated_2007}.
\section{Hochschild cohomology of categories}\label{Hochschild_cohomology_of_categories}
We work over a ground commutative ring $k$ and graded objects are $\mathbb Z$-graded. The degree of $x$ is denoted by $\abs{x}$. Let $\modules{k}$
be the category of graded $k$-modules equipped with the usual closed symmetric monoidal structure, where the symmetry constraint uses the Koszul sign rule. The tensor product will be denoted by $\otimes$ and the inner $\hom$ by $\hom_k^*$. Since we will not change rings, $k$ will often be dropped from notation.
A \emph{graded $k$-linear category}, or just \emph{graded category}, is a category enriched in $\modules{k}$, and similarly graded functors and graded natural transformations. Note that graded natural transformations may have any degree $n\in\mathbb Z$, while graded functors must be defined by degree $0$ morphisms between their graded modules of morphisms. We refer to \cite{kelly_basic_2005} for enriched category concepts. For the time being, all categories will be linear, so we will often drop this word.
Given a small graded category $\C C$, a \emph{right $\C C$-module} $M$ is a graded functor \[M\colon \C C^{\op}\To \modules{k}\] from the opposite graded category $\C C^{\op}$ to the category $\modules{k}$ of graded $k$-modules, i.e.~for each object $X$ in $\C C$ a graded module $M(X)$ is given, and for each pair of objects $X,X'$ in $\C C$ a degree $0$ morphism of graded modules
\begin{align*}
M(X)\otimes\C C(X',X)&\longrightarrow M(X'),\\
x\otimes f&\;\mapsto\; x\cdot f,
\end{align*}
satisfying the obvious associativity and unit conditions. A \emph{morphism} of right $\C C$-modules $g\colon M\rightarrow N$ of any degree $n\in\mathbb Z$ is a collection of degree $n$ morphims of graded $k$-modules $g(X)\colon M(X)\rightarrow N(X)$, $X$ an object in $\C C$, such that, with the notation above, $g(X')(x\cdot f)=g(X)(x)\cdot f$.
Right $\C C$-modules form a graded abelian category $\modules{\C C}$. Representable functors $\C C(-,X)$ and their shifts form a set of projective generators. A right $\C C$-module is \emph{finitely presented} if it is the cokernel of a morphism between finite direct sums of these projective generators. The full subcategory of finitely presented right $\C C$-modules will be denoted by $\modulesfp{\C C}$. Morphism graded $k$-modules in $\modules{\C C}$ are denoted by $\hom_{\C C}^\ast$ and their derived functors by $\ext^{n,\ast}_{\C C}$, $n\geq 0$. A good reference for this kind of graded categorical algebra is Street's thesis \cite{street_homotopy_1969}.
A \emph{bimodule} $M$ over $\C C$ is a right $\C C^{\env}$-module, where $\C C^{\env}=\C C\otimes \C C^{\op}$, i.e.~a graded functor
\[M\colon \C C^{\env}=\C C^{\op}\otimes\C C\To \modules{k}.\]
Equivalently, $M$ is a family of graded modules $M(X,Y)$ indexed by pairs of objects $X,Y$ in $\C C$
and, for each four objects $X,X',Y,Y'$ in $\C C$, a graded module morphism of degree $0$
\begin{align*}
\C C(Y,Y')\otimes M(X,Y)\otimes \C C(X',X)&\longrightarrow M(X',Y'),\\
g\otimes x\otimes f&\;\mapsto\; g\cdot x\cdot f,
\end{align*}
satisfying the usual associativity and unit conditions. The $\C C$-bimodule $\C C(-,-)$ will be simply denoted by $\C C$. A \emph{morphism} of $\C C$-bimodules $h\colon M\rightarrow N$ is a family of morphims of graded $k$-modules $h(X,Y)\colon M(X,Y)\rightarrow N(X,Y)$ satisfying \[h(X',Y')(g\cdot x\cdot f)=(-1)^{\abs{g}\abs{h}}g\cdot h(X,Y)(x)\cdot f.\]
The \emph{bar complex} $B_\star(\C C)$ is the chain complex of $\C C$-bimodules concentrated in dimensions $\geq 0$ defined by
\[B_n(\C C)=\bigoplus_{X_0,\dots, X_n} \C C(X_0,-)\otimes\C C(X_1,X_{0})\otimes\cdots\otimes\C C(X_n,X_{n-1})\otimes\C C(-,X_n),\]
where the coproduct is indexed by the sequences of $n+1$ objects of $\C C$,
with differential
\[d(f_0\otimes\dots\otimes f_{n+1})=\sum_{i=0}^n(-1)^i\cdots\otimes f_if_{i+1}\otimes\cdots.\]
It is a relative projective resolution of $\C C$ (with respect to $k$-split surjections) with augmentation
\[\epsilon\colon B_\star(\C C)\To\C C,\qquad\epsilon(f_0\otimes f_1)=f_0f_1.\]
If $\C C$ is locally projective, i.e.~if morphism graded modules in $\C C$ are projective, then it is an honest resolution.
We refer to \cite{mac_lane_homology_1963} for basic relative homological algebra.
The \emph{Hochschild cohomology} $\hh{\star,*}{\C C,M}$ of $\C C$ with coefficients in a $\C C$-bimodule $M$ is the cohomology of this complex (we should maybe say \emph{Hochschild--Mitchell cohomology} \cite{mitchell_rings_1972}). Using adjunction and the Yoneda lemma, the \emph{Hochschild cochain complex} $\hc{\star,*}{\C C,M}=\hom_{\C C^{\env}}^*(B_\star(\C C),M)$ is given in Hochschild degree $\star=n$ by
\[\hc{n,*}{\C C,M}=\prod_{X_0,\dots, X_n} \hom_{k}^*(\C C(X_1,X_{0})\otimes\cdots\otimes\C C(X_n,X_{n-1}),M(X_n,X_0)),\]
and the differential is given by
\begin{multline}\label{hochschild_differential}
d(\varphi)(f_1,\dots,f_{n+1})={}(-1)^{\abs{\varphi}\abs{f_1}}f_1\cdot \varphi(f_2,\dots,f_{n+1})\\+\sum_{i=1}^n(-1)^i \varphi(\dots, f_if_{i+1},\dots)+(-1)^{n+1}\varphi(f_1,\dots,f_n)\cdot f_{n+1}.
\end{multline}
If $M$ is a monoid in the category of $\C C$-bimoules, then Hochschild cohomology becomes a bigraded ring with the well-known cup-product.
Note that $\hh{0,*}{\C C,M}$ coincides in general with the \emph{end} of the bifunctor $M$.
Hochschild cohomology is functorial in both $\C C$ and $M$. A graded functor $F\colon \C D\rightarrow \C C$ and a $\C C$-bimodule morphism $\varphi\colon N\rightarrow M$ induce morphisms
\begin{align*}
F^*\colon \hh{\star,\ast}{\C C,M}&\To \hh{\star,\ast}{\C D,M(F,F)},\\
\varphi_*\colon \hh{\star,\ast}{\C C,N}&\To \hh{\star,\ast+\abs{\varphi}}{\C C,M}.
\end{align*}
They satisfy $F^*\varphi_*=\varphi(F,F)_*F^*$.
The bivariant functoriality can be described as in \cite{muro_functoriality_2006}, in particular categorical equivalences induce isomorphisms.
We consider the ungraded case as the particular instance of the former where everything is concentrated in degree $0$. Over ungraded categories, we can consider both graded and ungraded (bi)modules. If $\C T$ is an ungraded category and $M$ is a graded $\C T$-bimodule, then the cohomology of $\C T$ with coefficients in $M$ reduces to ungraded cohomology, \[\hh{p,q}{\C T,M}=\hh{p}{\C T,M^q}.\]
We now relate the cohomology of graded and ungraded categories.
We have sketched in the introduction how out of an arbitrary ungraded category $\C T$ equipped with an automorphism $\Sigma$, we can form a graded category $\C T_{\Sigma}$.
Composition in $\C T_\Sigma$ is defined as follows,
\begin{align*}
\C T_{\Sigma}^p(Y,Z)\otimes \C T_{\Sigma}^q(X,Y)&\To\C T_\Sigma^{p+q}(X,Z),\\
f\otimes g&\;\mapsto\; (\Sigma^qf)g.
\end{align*}
Here, on the right, we have a composition in $\C T$.
We can extend $\Sigma$ to an automorphism $\Sigma \colon\C T_\Sigma\rightarrow\C T_\Sigma$, defined as in $\C T$ on objects, and on morphisms as $(-1)^n\Sigma$ in each degree $n\in\mathbb Z$. In this way, the graded extension $\Sigma$ is equipped with a natural isomorphism
\begin{equation}\label{imath}
\imath_X\colon X\cong \Sigma X
\end{equation} of degree $-1$ given by the identity in $X$. The sign in the extension of $\Sigma$ is necessary for the graded naturality, because of Koszul's sign rule. Graded categories equivalent to some $\C T_\Sigma$ are called \emph{weakly stable} \cite{street_homotopy_1969}. They are characterized by the fact that shifts of representable functors are representable, or equivalently, each object has an isomorphism of any given degree.
Similarly, if $M$ is an ungraded $\C T$-bimodule equipped with an isomorphism
$\tau\colon M\cong M(\Sigma,\Sigma)$ such that, given
$g\in\C T(Y,Y')$, $x\in M(X,Y)$, and $f\in\C T(X',X)$,
\[\tau(g\cdot x\cdot f)=(\Sigma g)\cdot\tau(x)\cdot(\Sigma f),\]
then we can form a graded $\C T_{\Sigma}$-bimodule $M_\tau$ defined by
\[M_\tau^n(X,Y)=M(X,\Sigma^nY).\]
The bimodule structure is defined as
\begin{align*}
\C T^p_\Sigma(Y,Y')\otimes M^q_\tau(X,Y)\otimes \C T^r_\Sigma (X',X)
&\To
M^{p+q+r}_\tau(X',Y'),\\
g\otimes x\otimes f&\;\mapsto\;(\Sigma^{q+r}g)\cdot (\tau^rx)\cdot f.
\end{align*}
Here, on the right, we use the suspension in $\C T$ and the $\C T$-bimodule structure of $M$ (no signs involved). Moreover, we can extend $\tau$ to a degree $0$ isomorphism of $\C T_\Sigma$-bimodules $\tau\colon M_\tau\cong M_\tau(\Sigma,\Sigma)$ defined as $(-1)^n\tau$ in each degree $n\in\mathbb Z$. The sign in the definition of $\tau$ is needed to cancel the signs in the $\C T_\Sigma$-bimodule morphism equation for $\tau$ arising from the definition of the suspension in $\C T_\Sigma$.
Since $\C T\subset \C T_{\Sigma}$ is the degree $0$ part, we can also regard $M_\tau$ as a graded $\C T$-bimodule.
\begin{proposition}\label{graded_ungraded_long_exact_sequence}
If $\C T$ is an ungraded category equipped with an automorphism $\Sigma\colon \C T\rightarrow\C T$, $M$ is an ungraded $\C T$-bimodule equipped with an isomorphism $\tau\colon M\cong M(\Sigma,\Sigma)$ satisfying $\tau(f\cdot x\cdot g)=(\Sigma f)\cdot\tau(x)\cdot(\Sigma g)$, and $i\colon\C T\subset\C T_{\Sigma}$ denotes the inclusion of the degree $0$ part, then there is a long exact sequence
\begin{center}
\begin{tikzcd}[row sep=5mm]
\vdots\arrow[d]\\
\hh{n,*}{\C T_{\Sigma},M_{\tau}}\arrow[d, "i^*"]\\
\hh{n,*}{\C T,M_{\tau}}\arrow[d, "\operatorname{id}-\tau_*^{-1}\Sigma^*"]\\
\hh{n,*}{\C T,M_{\tau}}\arrow[d]\\
\hh{n+1,*}{\C T_{\Sigma},M_{\tau}}\arrow[d]\\
\vdots
\end{tikzcd}
\end{center}
\end{proposition}
\begin{proof}
We consider the $\C T_\Sigma$-bimodule $\C T_\Sigma\otimes_{\C T}\C T_\Sigma$. It is degreewise given by
\begin{equation*}
\begin{split}
(\C T_\Sigma\otimes_{\C T}\C T_\Sigma)^n&= \bigoplus_{p+q=n}\C T_\Sigma^p\otimes_{\C T}\C T_\Sigma^q\\
&=\bigoplus_{p+q=n}\C T(-,\Sigma^p)\otimes_{\C T}\C T(-,\Sigma^q)\\
&=\bigoplus_{q\in\mathbb Z}\C T(-,\Sigma^{n-q})\otimes_{\C T}\C T(-,\Sigma^q)\\
{\scriptstyle (\bigoplus_{q\in\mathbb Z}\Sigma^q\otimes 1)}&\cong
\bigoplus_{q\in\mathbb Z}\C T(\Sigma^q,\Sigma^n)\otimes_{\C T}\C T(-,\Sigma^q)\\
{\scriptstyle (\text{composition})}&\cong \bigoplus_{q\in\mathbb Z}\C T(-,\Sigma^n).
\end{split}
\end{equation*}
The natural transformation \eqref{imath} defines a degree $0$ bimodule automorphism
\[\Gamma\colon \C T_\Sigma\otimes_{\C T}\C T_\Sigma\cong \C T_\Sigma\otimes_{\C T}\C T_\Sigma\colon
f\otimes g\mapsto f\imath^{-1}\otimes\imath g.\]
Indeed, the left (resp.~right) tensor coordinate shifts its degree by $+1$ (resp.~$-1$) so, as a whole, it has degree $0$.
Degreewise, $\Gamma$ is the automorphism of $\bigoplus_{q\in\mathbb Z}\C T(-,\Sigma^n)$ which shifts coordinates one step downwards. Therefore, the degree $0$ bimodule morphism
\[\operatorname{id}-\Gamma\colon \C T_\Sigma\otimes_{\C T}\C T_\Sigma\longrightarrow \C T_\Sigma\otimes_{\C T}\C T_\Sigma \]
is injective (its kernel would be, degreewise, the elements whose coordinates are all equal to the previous coordinate, hence zero since we are in a direct sum).
Moreover, the bimodule morphism defined by composition
\[\C T_\Sigma\otimes_{\C T}\C T_\Sigma\longrightarrow \C T_\Sigma\colon f\otimes g\mapsto fg\]
is degrewise given by the identity in $\C T(-,\Sigma^n)$ on each direct summand, so it is the cokernel of $\operatorname{id}-\Gamma$, since $\operatorname{id}-\Gamma$ is actually the standard presentation of the colimit of the $\mathbb Z$-indexed diagram given by the identity in $\C T(-,\Sigma^n)$ everywhere. Suming up, we have a short exact sequence of $\C T_\Sigma$-bimodules
\[\C T_\Sigma\otimes_{\C T}\C T_\Sigma\stackrel{\operatorname{id}-\Gamma}\hookrightarrow \C T_\Sigma\otimes_{\C T}\C T_\Sigma\stackrel{\text{comp.}}\twoheadrightarrow \C T_\Sigma.\]
We will use it to construct a convenient $\C T_\Sigma$-bimodule resolution of $\C T_\Sigma$ from a resolution of $\C T_\Sigma\otimes_{\C T}\C T_\Sigma$ and a lift of $\operatorname{id}-\Gamma$.
The extension of scalars of the bar complex of $\C T$-bimodules $B_*(\C T)$ along the inclusion $i$ is $\C T_{\Sigma}\otimes_{\C T} B_*(\C T)\otimes_{\C T} \C T_{\Sigma}$, which is a relative projective resolution of $\C T_{\Sigma}\otimes_{\C T} \C T_{\Sigma}$. At each bar degree it is given by
\begin{multline*}
\C T_{\Sigma}\otimes_{\C T} B_n(\C T)\otimes_{\C T} \C T_{\Sigma}\\=\bigoplus_{X_0,\dots, X_n} \C T_{\Sigma}(X_0,-)\otimes\C T(X_1,X_{0})\otimes\cdots\otimes\C T(X_n,X_{n-1})\otimes\C T_\Sigma(-,X_n).
\end{multline*}
By adjunction, we can use this complex to compute the cohomology of $\C T$ with coefficients in the restriction of a $\C T_\Sigma$-bimodule (e.g.~$M_\tau$) along $i$.
The automorphism $\Gamma$ lifts to the bar resolution
\[\Gamma\colon \C T_{\Sigma}\otimes_{\C T} B_*(\C T)\otimes_{\C T} \C T_{\Sigma}\cong \C T_{\Sigma}\otimes_{\C T} B_*(\C T)\otimes_{\C T} \C T_{\Sigma}\]
by means of the $\C T_\Sigma$-bimodule isomorphisms
\begin{multline*}
\C T_{\Sigma} (X_0,-)\otimes\C T(X_1,X_{0})\otimes\cdots\otimes\C T(X_n,X_{n-1})\otimes\C T_{\Sigma} (-,X_n)\\\cong
\C T_{\Sigma} (\Sigma X_0,-)\otimes\C T(\Sigma X_1,\Sigma X_{0})\otimes\cdots\otimes\C T(\Sigma X_n,\Sigma X_{n-1})\otimes\C T_{\Sigma} (-,\Sigma X_n)
\end{multline*}
defined by
\begin{align*}
f_0\otimes f_1\otimes\cdots \otimes f_n\otimes f_{n+1}&\mapsto f_0\imath^{-1} \otimes (\Sigma f_1)\otimes\cdots\otimes(\Sigma f_n)\otimes \imath f_{n+1}.
\end{align*}
The mapping cone of $\operatorname{id}-\Gamma$ is therefore a relative projective resolution of $\C T_\Sigma$ as a bimodule over itself, which can be used to compute the cohomology of this category. The exact sequence in the statement will be the long exact cohomology sequence of the standard exact triangle completion of $\operatorname{id}-\Gamma$ with coefficients in $M_\tau$. It is only left to identify the morphism induced by $\Gamma$ on cohomology with $\tau^{-1}_*\Sigma^*$. We do this in the following paragraph, actually at the level of cochains.
A $\C T_\Sigma$-bimodule morphism of degree $m$
\[\varphi\colon \C T_{\Sigma}\otimes_{\C T} B_n(\C T)\otimes_{\C T} \C T_{\Sigma}\longrightarrow M_{\tau}\]
identifies with a collection of $k$-module morphisms
\[\varphi\colon \C T(X_1,X_{0})\otimes\cdots\otimes\C T(X_n,X_{n-1})\longrightarrow M_\tau^m(X_n,X_0)=M(X_n,\Sigma^mX_0),\]
one for each sequence of objects $X_0,\dots, X_n$. The composite $\varphi\Gamma$ is given by
\begin{equation*}
\begin{split}
\varphi\Gamma(f_1\otimes\cdots \otimes f_n)&=(-1)^m\imath^{-1}_{\Sigma^mX_0}\varphi((\Sigma f_1)\otimes\cdots\otimes(\Sigma f_n))\imath_{X_n}\\
&=(-1)^m\tau^{|\imath_{X_n}|}\varphi((\Sigma f_1)\otimes\cdots\otimes(\Sigma f_n))\\
&=\tau^{-1}\varphi(\Sigma^{\otimes^n})(f_1\otimes\cdots \otimes f_n).
\end{split}
\end{equation*}
Here we use that $\imath$ has deegree $-1$ and the $f_i$'s have degree $0$, that $\imath$ is given by identity maps, and the definitions of the $\C T_\Sigma$-module $M_\tau$ and the graded $\tau$.
\end{proof}
\begin{remark}[Multiplicative properties]\label{multiplicative_properties}
In the context of the previous proposition, assume that $M$ is a monoid in the category of $\C T$-bimodules and $\tau$ is a monoid morphism. Then $M_\tau$ is a monoid in the category of $\C T_\Sigma$-bimodules with composition law defined as
\[M_{\tau}^p(Y,Z)\otimes M_{\tau}^q(X,Y)\To M_{\tau}^{p+q}(X,Z)\colon x\otimes y\mapsto \tau^q(x)y.\]
Here, on the right, we use the ungraded $\tau$ and composition in $M$. The graded $\tau$ defined above becomes automatically a morphism of monoids in $\C T_\Sigma$-bimodules. Therefore, not only $\Sigma^*$ but also $\tau_*$ is a bigraded ring morphism $\hh{\star,*}{\C T,M_{\tau}}\rightarrow \hh{\star,*}{\C T,M_{\tau}(\Sigma,\Sigma)}$. In particular, $\tau_*^{-1}\Sigma^*$ is a bigraded ring endomorphism of $\hh{\star,*}{\C T,M_{\tau}}$ and the kernel of $\operatorname{id}-\tau_*^{-1}\Sigma^*$ is a bigraded subring, since it coincides with the equalizer of $\tau_*^{-1}\Sigma^*$ and the identity map.
\end{remark}
\section{Heller's classification of triangulated structures}\label{heller_section}
Triangulated categories were introduced by Puppe \cite{puppe_formal_1962} and Verdier \cite{verdier_categories_1996} at about the same time (Verdier's thesis, although widely circulated, was only published three decades later). Puppe, however, did not consider Verdier's \emph{octahedral axiom}. Eventually, Verdier's axiomatic became standard, hence we refer to Puppe triangulated structures if we do not explicitly require Verdier's additional axiom. It is remarkable that, to this day, after failed attempts, no example of Puppe triangulated category is known where the octahedral axiom fails, although it is a common belief that the octahedral axiom does not follow from the rest.
Freyd \cite{freyd_stable_1966} showed that, if an additive category $\C T$ is admits a Puppe triangulated structure with suspension $\Sigma$, then the category of finitely presented \emph{ungraded} right $\C T$-modules $\modulesfp{\C T}$ is a \emph{Frobenius} abelian category, i.e.~it has enough projectives and injectives and both classes of objects coincide (it is the class of direct summands of representable functors).
Assume that idemponents split in $\C T$ and $\modulesfp{\C T}$ is Frobenius abelian. Heller \cite{heller_stable_1968} classified the set of Puppe triangulated structures on $\C T$ with suspension $\Sigma$ in the following way. The \emph{stable module category} $\modulesst{\C T}$ is the quotient of $\modulesfp{\C T}$ by the ideal of morphisms factoring through a representable. Morphism sets in this category are denoted by $\homst_{\C T}$. The stable module category is a triangulated category, its suspension functor is the \emph{cosyzygy} functor $S $, defined on objects by the choice of short exact sequences in $\modulesfp{\C T}$ of the form
\[M\stackrel{j_M}\hookrightarrow\C T(-,X_M)\stackrel{q_M}\twoheadrightarrow S M.\]
Given a morphism $\{f\}\colon M\rightarrow N$ in $\modulesst{\C T}$ represented by $f$ in $\modulesfp{\C T}$, the morphism $S \{f\}$ is represented by any map $S f$ fitting in a commutative diagram
\begin{center}
\begin{tikzcd}
M \arrow[r, "j_M", hook] \arrow[d, "f"'] & \C T(-,X_M) \arrow[r, "p_M", two heads] \arrow[d] & S M \arrow[d, "S f"]\\
N \arrow[r, "j_N", hook] & \C T(-,X_N) \arrow[r, "p_N", two heads] & S N
\end{tikzcd}
\end{center}
(The class of exact triangles in $\modulesst{\C T}$ is irrelevant for our purposes.)
Moreover, the functor $\Sigma$ extends in an essentially unique way to an exact automorphism of $\modulesfp{\C T}$ through the Yoneda inclusion. Furthermore, it passes to the quotient $\modulesst{\C T}$ as a triangulated functor, part of which is a natural isomorphism \[\sigma\colon \Sigma S \cong S \Sigma\] defined by the choice of commutative diagrams in $\modulesfp{\C T}$ of the form
\begin{center}
\begin{tikzcd}
\Sigma M \arrow[r, "\Sigma j_M", hook] \arrow[d, equal] & \C T(-,\Sigma X_M) \arrow[r, "\Sigma p_M", two heads] \arrow[d] & \Sigma S M \arrow[d]\\
\Sigma M \arrow[r, "j_{\Sigma M}", hook] & \C T(-,X_{\Sigma M}) \arrow[r, "p_{\Sigma M}", two heads] & S \Sigma M
\end{tikzcd}
\end{center}
Heller \cite[Theorem 16.4]{heller_stable_1968} showed that the (possibly empty) set of Puppe triangulated structures on $\C T$ with suspension functor $\Sigma$ is in bijection with the set of natural isomorphisms \[\delta\colon\Sigma\cong S^{3}\] which anticommute with $S $, i.e.
\[(S \delta)\sigma+\delta S=0.\]
The set of natural transformations $\Sigma\rightarrow S^{3}$ is the cohomology group
\[\hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}\]
since $0$-dimensional Hochschild cohomology computes the end of the coefficient bimodule. Moreover, if we consider the natural isomorphism
\[\tau\colon\modulesst{\C T}(\Sigma,S^{3})\To
\modulesst{\C T}(\Sigma S ,S^{4})\colon\delta\mapsto -(S\delta)\sigma,\]
the anticommutativity condition is equivalent to being in the kernel of
\begin{equation}\label{anticommutativity_cohomological}
\operatorname{id}-\tau_*^{-1}S^*\colon \hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}\longrightarrow
\hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}.
\end{equation}
The pull-back of the $\modulesst{\C T}$-bimodule $\homst_{\C T}(-,S^n)$ along the natural projection $\modulesfp{\C T}\twoheadrightarrow\modulesst{\C T}$ is \emph{Tate}'s $\widehat{\ext}^{n}_{\C T}$, $n\in\mathbb Z$, computed by using a complete resolution of any variable instead of just a projective or an injective resolution. It coincides with the ordinary $\ext_{\C T}^n$ for $n>0$. Composition in $\modulesst{\C T}_S$ extends the Yoneda product in $\ext_{\C T}^*$. In order to compute the coend of a $\modulesst{\C T}$-bimodule, we can equally pull it back to $\modulesfp{\C T}$. Hence, combining the previous observations, we obtain an isomorphism
\[\hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}
\cong
\hh{0}{\modulesfp{\C T},\widehat{\ext}_{\C T}^3(-,\Sigma^{-1})}
.\]
Any cocycle $\varphi$ on the right gives rise to a class of exact triangles. More precisely, a diagram in $\C T$ of the form
\[X\stackrel{f}{\To}Y\stackrel{i}{\To}C_f\stackrel{q}{\To}\Sigma X\]
is a \emph{$\varphi$-exact triangle} if
\begin{center}
\begin{tikzcd}
\Sigma^{-1}M\arrow[r,"\alpha",hook] & \C T(-,X)\arrow[r,"{\C T(-,f)}"] &\C T(-,Y)\arrow[r, "{\C T(-,i)}"]&\C T(-,C_f)\arrow[r,"\lambda",two heads]&M,
\end{tikzcd}
\end{center}
where $M=\operatorname{Im} \C T(-,q)$ and
$\C T(-,q)=(\Sigma\alpha)\lambda$, is an extension representing $\varphi(M)\in\widehat{\ext}_{\C T}^3(M,\Sigma^{-1}M)$. Heller's result is equivalent to saying that this defines a bijection between Puppe triangulated stuctures in $\C T$ with suspension $\Sigma$ and Hochschild $0$-cocycles satisfying the invertibility and the anticommutativity conditions. So far, in order to check these conditions we need the natural transformation $\delta$. We will now give a cohomological characterization of the invertibility condition. The anticommutativity condition has already a characterization using the cohomology of $\modulesst{\C T}$, see \eqref{anticommutativity_cohomological}. We will later give a characterization in terms of the cohomology of $\modulesfp{\C T_\Sigma}$, in Section \ref{spectral_sequence_section} over a field, and in Section \ref{topological_section} in general.
Since $\modulesfp{\C T}$ is Frobenius abelian, then $\modulesfp{\C T_{\Sigma}}$ is graded Frobenius abelian. (The former is the degree $0$ part of the latter, and these properties are characterized in degree $0$.) Therefore, on finitely presented $\C T_{\Sigma}$-modules, we have Tate's $\widehat{\ext}_{\C T_{\Sigma}}^{n,*}$, $n\in\mathbb Z$, as above.
The invertible functor $\Sigma\colon\modulesfp{\C T}\rightarrow\modulesfp{\C T}$ induces natural isomorphisms
\[\Sigma\colon \widehat{\ext}_{\C T}^{n}\cong \widehat{\ext}_{\C T}^{n}(\Sigma,\Sigma)\]
compatible with the extended Yoneda product
for all $n$, also denoted by $\Sigma$.
It is easy to see that, using the notation of the previous section,
\[
\modulesfp{\C T}_\Sigma=\modulesfp{\C T_\Sigma},\qquad
(\widehat{\ext}_{\C T}^{n})_{\Sigma}\cong \widehat{\ext}_{\C T_\Sigma}^{n,*}.\]
The extension of the Yoneda product endows $\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}$ with a bigraded monoid structure in the category of $\modulesfp{\C T_{\Sigma}}$-bimodules, hence
\[\hh{\star}{\modulesfp{\C T},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}\]
is a trigraded ring. Its units are concentrated in Hochschild degree $\star=0$ since $\star\geq 0$, while $\bullet$ and $*$ may be arbitrary integers.
\begin{proposition}\label{Heller_ungraded}
If $\C T$ is a small ungraded idempotent complete additive category such that $\modulesfp{\C T}$ is Frobenius abelian and $\Sigma\colon \C T\rightarrow\C T$ is an automorphism, then the set of Puppe triangulated structures on $\C T$ with suspension functor $\Sigma$ is in bijection with the units of the bigraded ring $\hh{0}{\modulesfp{\C T},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$ lying in
\[\hh{0}{\modulesfp{\C T},\widehat{\ext}_{\C T_{\Sigma}}^{3,-1}}\cong \hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}\]
and satisfying Heller's anticommutativity condition.
\end{proposition}
\begin{proof}
Both the condition of a natural transformation $\Sigma\rightarrow S^3$ being invertible or the corresponding cocycle on the left being a unit can be translated in saying that the extension class $\varphi(M)$ associated to a finitely presented right $\C T$-module $M$ can be represented by an extension with representable (i.e.~projective-injective) middle terms
\[\Sigma^{-1}M\hookrightarrow\C T(-,X)\rightarrow\C T(-,Y)\rightarrow\C T(-,Z)\twoheadrightarrow M.\]
(In general, only either the two ones on the left or the two ones on the right can be taken to be representable.)
\end{proof}
This new glimpse at Heller's result allows to place the set of Puppe triangulated structures in a smaller recipient. First, note that Proposition \ref{graded_ungraded_long_exact_sequence} yields an exact sequence
\begin{equation}\label{exact_cohomology_sequence}
\begin{tikzcd}[row sep=5mm]
0\arrow[d, hook]\\
\hh{0,-1}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_\Sigma}^{3,*}}\arrow[d, "i^*"]\\
\hh{0,-1}{\modulesfp{\C T},\widehat{\ext}_{\C T_\Sigma}^{3,*}}\arrow[d, "\operatorname{id}-\Sigma^{-1}_*\Sigma^*"]\\
\hh{0,-1}{\modulesfp{\C T},\widehat{\ext}_{\C T_\Sigma}^{3,*}}
\end{tikzcd}
\end{equation}
where the bottom map coincides with
\begin{equation*}
\begin{tikzcd}[row sep=5mm]
\hh{0}{\modulesfp{\C T},{\ext}_{\C T}^{3}(-,\Sigma^{-1})}\arrow[d, "\operatorname{id}+\Sigma^{-1}_*\Sigma^*"]\\
\hh{0}{\modulesfp{\C T},{\ext}_{\C T}^{3}(-,\Sigma^{-1})}
\end{tikzcd}
\end{equation*}
The apparent change of sign in the morphism is motivated by the definition of the graded bimodule morphism $\Sigma$ on the coefficients.
We now consider the trigraded ring
\[\hh{\star,*}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}},\]
which, as above, only contains units in the Hochschild degree $\star=0$ subring.
\begin{corollary}\label{Heller_graded}
If $\C T$ is a small ungraded idempotent complete additive category such that $\modulesfp{\C T}$ is Frobenius abelian and $\Sigma\colon \C T\rightarrow\C T$ is an automorphism, then the set of Puppe triangulated structures on $\C T$ with suspension functor $\Sigma$ is in bijection with the units of the bigraded ring $\hh{0,*}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$ lying in
\[\hh{0,-1}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{3,*}}\] whose image along $i^*$ satisfies Heller's anticommutativity condition.
\end{corollary}
\begin{proof}
We are in the conditions of Remark \ref{multiplicative_properties}, hence $\hh{0,*}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$ is the equalizer of a pair of ring endomorphisms of $\hh{0,*}{\modulesfp{\C T},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$. This shows that the inclusion $i^*$ not only preserves but also reflects units.
It remains to check that any Heller element $\delta\in \hh{0}{\modulesfp{\C T},\widehat{\ext}_{\C T_{\Sigma}}^{3,-1}}$ is in the kernel of $\operatorname{id}-\Sigma^{-1}_*\Sigma^*$. By the cohomological characterization of Heller's anticommutativity condition, see \eqref{anticommutativity_cohomological}, $\delta$ is in the kernel of $\operatorname{id}-\tau^{-1}_*S^*$. Using the natural isomorphism $\Sigma\cong S^3$ provided by $\delta$, we have a square of functors commuting up to natural isomorphism and two commutative squares of bimodules as follows,
\begin{center}
\begin{tikzcd}
\modulesfp{\C T}\arrow[r,"\Sigma"]\arrow[d, two heads]&\modulesfp{\C T}\arrow[d, two heads]\\
\modulesst{\C T}\arrow[r,"S^3"]&\modulesst{\C T}
\end{tikzcd}
\qquad
\begin{tikzcd}
{\ext}_{\C T}^{3}(-,\Sigma^{-1})\arrow[r,"-\Sigma"]\arrow[d, "\cong"']&{\ext}_{\C T}^{3}(\Sigma,-)\arrow[d, "\cong"]\\
\widehat{\ext}_{\C T_\Sigma}^{3,-1}\arrow[r,"\Sigma"]\arrow[d, "\cong"']&\widehat{\ext}_{\C T_\Sigma}^{3,-1}(\Sigma,\Sigma)\arrow[d, "\cong"]\\
\homst_{\C T}(\Sigma, S^3)\arrow[r,"\tau^3"]&\homst_{\C T}(\Sigma S^3, S^6)
\end{tikzcd}
\end{center}
Hence we can identify $\operatorname{id}-\Sigma^{-1}_*\Sigma^*$ with $\operatorname{id}-\tau^{-3}_*(S^3)^*$. The result follows since the kernel of $\operatorname{id}-\tau^{-1}_*S^*$ is clearly contained in the kernel of $\operatorname{id}-\tau^{-3}_*(S^3)^*$.
\end{proof}
Since Heller's anticommutativity condition reduces to being in the kernel of a morphism, we can therefore think that the set of Puppe triangulated structures is a `locally closed' subset of the `affine space' $\hh{0,-1}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{3,*}}$. (This is literal in the cases where the latter is a finite dimensional vector space over some field $k$, and this happens under appropriate finiteness assumptions on a $k$-linear $\C T$.) We will later give a neater cohomological characterization of the anticommutativity condition in terms of the cohomology of $\modulesfp{\C T_\Sigma}$ alone, see Sections \ref{spectral_sequence_section} and \ref{topological_section}.
\section{Toda brackets}\label{toda_brackets_section}
Together with his classification of Puppe triangulated structures in terms of natural transformations, recalled in the previous section, Heller embedded these triangulated structures into a set of operations called Toda brackets \cite[Theorem 13.2]{heller_stable_1968}. In this section we see that our new cohomological approach to Heller's theory fits perfectly with the Toda bracket perspective.
\begin{definition}\label{Toda_bracket}
Let $\C{T}$ be an additive category and $\Sigma\colon\C{T}\rightarrow \C{T}$ an automorphism. A \emph{Toda bracket} $\langle -,-,-\rangle$ is an operation which sends three composable morphisms
\[X\st{f}\To Y\st{g}\To Z\st{h}\To T\]
with $g\cdot f=0$ and $h\cdot g=0$ to an element
\[\langle h,g,f\rangle
\in\frac{\C{T}( X,\Sigma^{-1}T)}{(\Sigma^{-1}h)\cdot\C{T}( X, \Sigma^{-1}Z)+\C{T}(Y,\Sigma^{-1}T)\cdot f},\]
often regarded as a subset $\langle h,g,f\rangle\subset \C{T}(X,\Sigma^{-1}T)$.
The following axioms must hold, whenever the Toda brackets are defined,
\begin{align*}
\langle i, h,g\rangle\cdot f&\subset\langle i, h,g\cdot f\rangle,& \langle i,h\cdot g, f\rangle&\supset\langle i\cdot h,g,f\rangle,\\
\langle i, h,g\cdot f\rangle&\subset\langle i, h\cdot g, f\rangle,&
\langle i\cdot h, g, f\rangle&\supset (\Sigma^{-1}i)\cdot \langle h,g,f\rangle.
\end{align*}
We say that a Toda bracket is \emph{stable} if in addition
\begin{align*}
\langle \Sigma h,\Sigma g,\Sigma f\rangle&=- \Sigma \langle h,g,f\rangle.
\end{align*}
The set of Toda brackets on $(\C T,\Sigma)$ is an abelian group with sum given by pointwise addition (even a $k$-module if our category is $k$-linear, since Toda brackets can be rescaled). It will be denoted by
\[TB(\C{T},\Sigma).\]
Stable Toda brackets form a subgroup (or submodule) denoted by
\[TB^s(\C{T},\Sigma).\]
\end{definition}
Usually, the recipient of a Toda bracket is equivalently taken to be the isomorphic group
\[\frac{\C{T}(\Sigma X,T)}{h\cdot\C{T}(\Sigma X, Z)+\C{T}(\Sigma Y,T)\cdot(\Sigma f)}.\]
Our convention here, however, fits better with the rest of this paper.
The well-known Toda bracket of a triangulated structure is stable, and it is determined by the previous laws and the fact that the Toda bracket of an exact triangle contains the identity. Conversely, this also defines the triangulated structure from the Toda bracket.
The last result of the previous section places the set of Puppe triangulated structures on $(\C T, \Sigma)$ within a graded Hochschild cohomology group, smaller than Heller's set of natural transformations, which has also been reinterpreted as a larger ungraded Hochschild cohomology group. In the following result we show that Heller's set of natural transformations is in bijection with the set of Toda brackets, and stable ones are in bijection with our smaller Hochschild cohomology group.
\begin{theorem}\label{Toda_brackets_and_cohomology}
Given an idempotent complete additive category $\C T$ such that the category $\modulesfp{\C T}$ is abelian and an automorphism $\Sigma\colon\C T\rightarrow\C T$, there are isomorphisms compatible with the inclusions,
\begin{align*}
\hh{0}{\modulesfp{\C T},\ext_{\C T}^3(-,\Sigma^{-1})}&\cong TB(\C{T},\Sigma),\\
\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}&\cong TB^s(\C{T},\Sigma).
\end{align*}
\end{theorem}
\begin{proof}
The second isomorphism follows from the first one together with Corollary \ref{graded_ungraded_long_exact_sequence} and the exact sequence \eqref{exact_cohomology_sequence} since under the first isomorphism, defined below, the stability condition for Toda brackets is equivalent to being in the kernel of $1-\Sigma_*^{-1}\Sigma^*$.
We now start with the first cohomological interpretation of Toda brackets. A sequence of maps in $\C T$
\[X\st{f}\To Y\st{g}\To Z\st{h}\To T\]
with $g\cdot f=0$ and $h\cdot g=0$ is the same as a chain complex $P_*$ of projectives in $\modulesfp{\C T}$ concentrated in degrees from $0$ to $3$, and the Toda bracket is an element $\varphi(P_*)\in H^3(P_*,\Sigma^{-1}H_0(P_*))$. Here and below $\Sigma$ is not the classical suspension of chain complexes, but the exact invertible endofunctor $\Sigma$ of $\modulesfp{\C T}$. The four properties a Toda bracket amount to saying that $\varphi(P_*)$ is natural with respect to chain maps $Q_*\rightarrow P_*$ between such complexes. Hence, $\varphi$ is the same as a cocycle in
\[\hh{0}{\chain_{\geq 0}^{\leq 3}(\C T),H^3(-,\Sigma^{-1}H_0(-))},\]
where $\chain_{\geq 0}^{\leq 3}(\C T)$ is the category of chain complexes as above. We will now define a sequence of isomorphisms
\begin{align*}
\hh{0}{\chain_{\geq 0}^{\leq 3}(\C T),H^3(-,\Sigma^{-1}H_0(-))}&\cong \hh{0}{\chain_{\geq 0}(\C T),H^3( w_{\leq 3}(-),\Sigma^{-1}H_0(-))}\\
&\cong \hh{0}{\chain_{\geq 0}(\C T),H^3(-,\Sigma^{-1}H_0(-))}\\
&\cong \hh{0}{\derived_{\geq 0}(\C T),
\derived_{\geq 0}(\C T)(-,\Sigma^{-1}H_0(-)[3])}\\
&\cong \hh{0}{\modulesfp{\C T},\ext_{\C T}^3(-,\Sigma^{-1})}.
\end{align*}
Here $\chain_{\geq 0}(\C T)$ is the category of non-negative chain complexes of projectives in $\modulesfp{\C T}$. It fits in an adjoint pair
\begin{center}
\begin{tikzcd}[column sep=30mm]
\chain_{\geq 0}^{\leq 3}(\C T)\arrow[r, shift left, "\text{inclusion}"]&\chain_{\geq 0}(\C T)\arrow[l, shift left, "\text{naive truncation }w_{\leq 3}"]
\end{tikzcd}
\end{center}
which yields the first isomorphism, induced by $w_{\geq 3}$, compare \cite[Theorem 5.10]{muro_functoriality_2006}.
The $\chain_{\geq 0}(\C T)$-bimodule $H^3(w_{\leq 3}(-),\Sigma^{-1}H_0(-))$ is
\begin{multline*}
H^3(w_{\leq 3}(P_*),\Sigma^{-1}H_0(Q_*))\\=\coker[\hom_{\C T}(P_2,\Sigma^{-1}H_0(Q_*))\rightarrow\hom_{\C T}(P_3,\Sigma^{-1}H_0(Q_*))],
\end{multline*}
hence we have a short exact sequence of $\chain_{\geq 0}(\C T)$-bimodules
\[H^3(-,\Sigma^{-1}H_0(-))\hookrightarrow H^3(w_{\leq 3}(-),\Sigma^{-1}H_0(-))\twoheadrightarrow M,\]
where $M$ is defined by the images
\[M(P_*,Q_*)=\im[\hom_{\C T}(P_3,\Sigma^{-1}H_0(Q_*))\rightarrow\hom_{\C T}(P_4,\Sigma^{-1}H_0(Q_*))].\]
We now check that
\[\hh{0}{\chain_{\geq 0}(\C T),M}=0,\]
so the second isomorphism follows from the long exact cohomology sequence associated to the previous short exact sequence of coefficient bimodules. Indeed, given a cocycle $\psi\in \hh{0}{\chain_{\geq 0}(\C T),M}$ and an object $P_*$ in $\chain_{\geq 0}(\C T)$, we can consider the complex $Q_*$ which reduces to $P_4$ in degrees $3$ and $4$ (the only possibly non-trivial differential being the identity) and the only map $f\colon Q_*\rightarrow P_*$ which is the identity in degree $4$. The cocycle condition says that
\[f\cdot \psi(Q_*)=\psi(P_*)\cdot f\in M(Q_*,P_*)=\hom_{\C T}(P_4,\Sigma^{-1}H_0(P_*)).\]
Moreover, the map $M(P_*,P_*)\rightarrow M(Q_*,P_*)$ defined by right multiplication by $f$ is the inclusion of the image, hence injective, and $\psi(Q_*)=0$ since $H_0(Q_*)=0$, therefore $\psi(P_*)=0$, so $\psi=0$.
The dervied category $\derived_{\geq 0}(\C T)$ of non-negative chain complexes of finitely presented $\C T$-modules comes equipped with a canonical functor $\chain_{\geq 0}(\C T)\rightarrow \derived_{\geq 0}(\C T)$. The pull-back along this functor of the
bimodule
$\derived_{\geq 0}(\C T)(-,\Sigma^{-1}H_0(-)[3])$, where $[3]$ denotes the $3$-fold shift in the derived category, is $H^3(-,\Sigma^{-1}H_0(-))$. Moreover, the canonical functor factors through an equivalence from the (quotient) homotopy category of the source to the target. Hochschild cohomology computes ends,
and they can be equally computed in $\chain_{\geq 0}(\C T)$ or in the `quotient' $\derived_{\geq 0}(\C T)$. Hence we obtain the third isomorphism, induced by the previous canonical functor.
We now consider the adjunction
\begin{center}
\begin{tikzcd}[column sep=30mm]
\derived_{\geq 0}(\C T)\arrow[r, shift left, "H_0"]&\modulesfp{\C T}.\arrow[l, shift left, "\text{inclusion in degree }0"]
\end{tikzcd}
\end{center}
It is actually a reflection. The left $\derived_{\geq 0}(\C T)$-module structure on the coefficient bimodule
$\derived_{\geq 0}(\C T)(-,\Sigma^{-1}H_0(-)[3])$ factors through $H_0$, hence one can check as in \cite[Theorem 5.4]{muro_functoriality_2006} that the degree $0$ inclusion induces an isomorphism in cohomology, the last one above.
Unwrapping the previous isomorphisms, the Toda bracket associated to a cocycle $\varphi\in \hh{0}{\modulesfp{\C T},\ext_{\C T}^3(-,\Sigma^{-1})}$ can be computed as follows. Let
\[X\st{f}\To Y\st{g}\To Z\st{h}\To T\]
be a sequence of maps in $\C T$
with $g\cdot f=0$ and $h\cdot g=0$. We pick up a projective resolution $\C T(-,U_*)$ of $M=\coker\C T(-,h)$ in $\modulesfp{\C T}$. By standard homological algebra, there is a map of complexes, unique up to chain homotopy,
\begin{center}
\begin{tikzcd}[column sep=6.5mm]
\cdots\arrow[r]&
0\arrow[r]\arrow[d]&
\C T(-,X)\arrow[r,"{\C T(-,f)}"]\arrow[d]&
\C T(-,Y)\arrow[r, "{\C T(-,g)}"]\arrow[d]&
\C T(-,Z)\arrow[r, "{\C T(-,h)}"]\arrow[d]&
\C T(-,T)\arrow[d]\\
\cdots\arrow[r]&
\C T(-,U_4)\arrow[r]&
\C T(-,U_3)\arrow[r]&
\C T(-,U_2)\arrow[r]&
\C T(-,U_1)\arrow[r]&
\C T(-,U_0)&
\end{tikzcd}
\end{center}
which induces the identity in $0$-dimensional homology (it is $M$ in both cases). This chain map induces a morphism in $3$-dimensional cohomology with coefficients in $\Sigma^{-1}M$,
\[\ext_{\C T}^3(M,\Sigma^{-1}M)\To
\frac{\C{T}( X,\Sigma^{-1}T)}{(\Sigma^{-1}h)\cdot\C{T}( X, \Sigma^{-1}Z)+\C{T}(Y,\Sigma^{-1}T)\cdot f}.\]
The Toda bracket $\langle f,g,h\rangle$ is the image of $\varphi(M)$ along this morphism.
\end{proof}
\section{A local-to-global spectral sequence}\label{spectral_sequence_section}
We now construct the spectral sequence which defines the first obstructions for the existence of an enhancement of a triangulated category over a field.
\begin{proposition}\label{spectral_sequence}
If $\C C$ is a small graded category over a field $k$, there is a first quadrant cohomological spectral sequence of graded $k$-modules
\[E_2^{p,q}=\hh{p,*}{\modulesfp{\C C},\ext_{\C C}^{p,*}}\Longrightarrow\hh{p+q,*}{\C C,\C C}.\]
\end{proposition}
\begin{proof}
The spectral sequence will be associated to the bicomplex
\[C^{\star,\bullet}=\hom_{\modulesfp{\C C}^{\env}}^*(B_{\star}(\modulesfp{\C C}),\hom_{\C C}^*(-\otimes_{\C C}B_{\bullet}(\C C),-)).\]
An element of $C^{p,q}$ is the same a family of graded $k$-module morphisms
\[\bigotimes_{i=1}^p\hom^*_{\C C}(M_{i},M_{j-1})
\otimes M_p(X_0)\otimes
\bigotimes_{j=1}^q\C C(X_{j},X_{j-1})\longrightarrow M_0(X_q)\]
indexed by all sequences of objects $M_0,\dots, M_p$ in $\modulesfp{\C C}$ and $X_0,\dots, X_q$ in $\C C$, and, with this description, the horizontal and vertical differentials are
\begin{equation}\label{differentials_bicomplex}
\begin{split}
d_h(\varphi)(g_1,\dots, g_{p+1},x,f_1,\dots, f_q)={}&(-1)^{\abs{\varphi}\abs{g_1}}g_1(X_q)(\varphi(g_2,\dots, g_{p+1},x,f_1,\dots, f_q))\\
&+\sum_{i=1}^p(-1)^i\varphi(\dots,g_ig_{i+1},\dots,x,f_1,\dots, f_q)\\
&+(-1)^{p+1}\varphi(g_1,\dots, g_{p+1}(X_0)(x),f_1,\dots, f_q),\\
d_v(\varphi)(g_1,\dots, g_{p},x,f_1,\dots, f_{q+1})={}&\varphi(g_1,\dots, g_{p},x\cdot f_1,\dots, f_{q+1})\\
&+\sum_{i=1}^q(-1)^i\varphi(g_1,\dots, g_{p},x,\dots, f_if_{i+1},\dots)\\
&+(-1)^{q+1}\varphi(g_1,\dots, g_{p},x,f_1,\dots, f_q)\cdot f_{q+1}.
\end{split}
\end{equation}
Since we are working over a field, all categories are \emph{locally free}, i.e.~morphism graded modules are free, and bar resolutions are honest projective bimodule resolutions. Moreover, for any right $\C C$-module $M$, the complex of right $\C C$-modules $M\otimes_{\C C}B_{\bullet}(\C C)$ is a projective resolution. Here we use that augmented bar resolutions, in general, admit a contraction as complexes of left or right modules (not as complexes of bimodules), so the homology of $M\otimes_{\C C}B_{\bullet}(\C C)$ is $M$ is concentrated in degree $0$. We also use that
\[M\otimes_{\C C}B_{n}(\C C)
=
\bigoplus_{X_0,\dots, X_n} M(X_0)\otimes\C C(X_1,X_{0})\otimes\cdots\otimes\C C(X_n,X_{n-1})\otimes\C C(-,X_n)\]
and $M$ takes free values since the ground ring is a field, so this is a direct sum of representables, hence projective.
Since $\modulesfp{\C C}$ is locally free $B_{\star}(\modulesfp{\C C})$ is a \emph{projective} resolution of $\modulesfp{\C C}$ as a bimodule over itself, so the $E_2$-term of the first-vertical-then-horizontal cohomology spectral sequence is
\[E_2^{p,q}=\hh{p,*}{\modulesfp{\C C},\ext_{\C C}^{p,*}}.\]
In order to compute the target of this spectral sequence, i.e.~the cohomology of the total complex of $C^{\star,\bullet}$, we now look at the $E_2$-term of the other spectral sequence associated to the bicomplex.
The $\modulesfp{\C C}$-bimodule $\hom_{\C C}^*(-\otimes_{\C C}B_{n}(\C C),-)$ sends $M$ and $N$ to
\[\prod_{X_1,\dots,X_n}\hom^*_k(M(X_0)\otimes\C C(X_1,X_{0})\otimes\cdots\otimes\C C(X_n,X_{n-1}),N(X_n)),\]
hence it is a product of $\modulesfp{\C C}$-bimoules $D_{X,Y}$ of the form \[D_{X,Y}(M,N)=\hom_{k}^*(M(X),N(Y)),\] where $X$ and $Y$ are fixed objects in $\C C$. Here we use again that we are working over a field, so $k$-module morphism objects in $\C C$ are free. The cohomology of $\modulesfp{\C C}$ with coefficients in such a $D_{X,Y}$ vanishes in positive dimensions. For this, we use the chain homotopy $h$ defined on
$\hom_{\C C^{\env}}^*(B_\bullet(\modulesfp{\C C}),D_{X,Y})$ as follows, compare \cite[Lemma 3.10]{jibladze_cohomology_1991}. Given an $(n+1)$-cochain $\varphi$ and morphisms $f_i\colon M_i\rightarrow M_{i-1}$ in $\modulesfp{\C C}$, $n\geq 0$, $1\leq i\leq n$, $h(\varphi)(f_1\otimes \cdots\otimes f_n)\in D_{X,Y}(M_n,M_0)$ is defined by
\begin{align*}
h(\varphi)(f_1\otimes \cdots\otimes f_n)\colon \hom_{\C C}^*(\C C(-,X),M_n)&=M_n(X)\To M_0(Y),\\
g&\longmapsto \varphi(f_1\otimes \cdots\otimes f_n\otimes g)(1_X).
\end{align*}
Hence, the inclusion of the $0$-dimensional horizontal cohomology in $C^{\star,\bullet}$ is a quasi-isomorphism (we mean with the total complex of $C^{\star,\bullet}$). This $0$-dimensional horizontal cohomology is the end of the cochain complex of $\modulesfp{\C C}$-bimodules $\hom_{\C C}^*(-\otimes_{\C C}B_{\bullet}(\C C),-)$. Such end is, dimension-wise, the graded module of graded natural transformations from the source to the target regarded as graded functors $\modulesfp{\C C}\rightarrow \modules{\C C}$. The source preserves colimits, and $\C C\subset\modulesfp{\C C}$ is the inclusion of a dense subcategory \cite[\S5.1]{kelly_basic_2005}, hence the source is the left Kan extension of its restriction along $\C C\subset\modulesfp{\C C}$ \cite[Theorem 5.29]{kelly_basic_2005}, so the end can be computed by restricting to $\C C$. The latter end is the complex $\hom_{\C C^{\env}}^*(B_\bullet(\C C),\C C)$, whose cohomology is the claimed target of the spectral sequence.
The explicit quasi-isomorphism $\xi\colon \hom_{\C C^{\env}}^*(B_\bullet(\C C),\C C)\hookrightarrow C^{\star,\bullet}$ is defined as follows. Given an $n$-cochain $\varphi$ in the source, a finitely presented $\C C$-bimodule $M$, morphisms $f_i\colon X_i\rightarrow X_{i-1}$ in $\C C$, $n\geq 0$, $1\leq i\leq n$, and $g\in M(X_0)$,
\begin{equation}\label{quasi_iso}
\xi(\varphi)(M)(g, f_1,\cdots, f_n)=(-1)^{\abs{\varphi}\abs{g}}g\varphi(f_1,\cdots, f_n).
\end{equation}
\end{proof}
\begin{remark}
In the previous proposition we can replace $\modulesfp{\C C}$ with any small full subcategory $\C B\subset\modules{\C C}$ containing $\C C$ since then $\C C$ is dense in $\C B$.
The spectral sequence has an ungraded version, where $\C C$ is an ungraded category and $\modulesfp{\C C}$ is replaced with the category of finitely presented ungraded right $\C C$-modules, or any full subcategory $\C B$ of the category of ungraded $\C C$-modules containing $\C C$. The proof is exactly the same.
The ungraded spectral sequence has been considered in \cite[Theorem 5.4.1]{lowen_hochschild_2005}. There, $\C C$ is replaced with the category of injective objects in an Grothendieck abelian category, but this is essentially equivalent to our framework because the Hochschild cohomology of a category coincides with that of its opposite.
\end{remark}
We now prove that, for a particular instance of the previous spectral sequence, the edge morphism takes the universal Massey product of an $A$-infinity model of a triangulated category to the Toda bracket which classifies the triangulated structure.
\begin{theorem}\label{theorem_edge_morphism}
Let $\C T$ be an idempotent complete triangulated category over a field $k$ with suspension $\Sigma$. Assume we have a minimal $A$-infinity enhancement with universal Massey product $\{m_3\}\in \hh{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}$. Then the edge morphism
\[\hh{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\To \hh{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}
\]
of the spectral sequence in Proposition \ref{spectral_sequence} for $\C C=\C T_{\Sigma}$ takes the universal Massey product to the Toda bracket of the triangulated structure.
\end{theorem}
\begin{proof}
Given an exact triangle
\[X\stackrel{f}{\To}Y\stackrel{i}{\To}C_f\stackrel{q}{\To}\Sigma X\]
in $\C T$,
\begin{center}
\begin{tikzcd}[column sep=5mm]
\cdots\arrow[r]&
\C T(-,\Sigma^{-1}C_f)\arrow[r, "{\C T(-,\Sigma^{-1}q)}" yshift=4pt]\arrow[r] &
\C T(-,X)\arrow[r,"{\C T(-,f)}" yshift=4pt] &
\C T(-,Y)\arrow[r, "{\C T(-,i)}" yshift=4pt] &
\C T(-,C_f)\arrow[r, "{\C T(-,q)}" yshift=4pt] &
\C T(-,\Sigma X)
\end{tikzcd}
\end{center}
is a projective resolution of $\coker\C T(-,q)$ in $\modulesfp{\C T}$.
Combining this fact with the explicit formulas in the proofs of Theorem \ref{Toda_brackets_and_cohomology} and Proposition \ref{spectral_sequence}, we see that the Toda bracket defined by the image of $\{m_3\}$ along the edge morphism in the statement and the isomorphism in Theorem \ref{Toda_brackets_and_cohomology} satisfies
\[m_3(h,g,f)\in\langle h,g,f\rangle.\]
By \cite{bondal_enhanced_1991}, this is the Toda bracket associated to the triangulated structure of $\C T$ with suspension $\Sigma$, since it is enhanced by the minimal $A$-infinity category structure with universal Massey product $\{m_3\}$.
\end{proof}
Next, we define certain morphisms within the $E_2$-term of the previous spectral sequence. These morphisms will be used afterwards to give a new cohomological interpretation of Heller's anticommutativity condition.
\begin{definition}\label{kappa}
Let $\C C$ be a graded category such that $\modulesfp{\C C}$ is graded Frobenius abelian. We define the graded $k$-module morphisms, $p\geq 1$, $q\in\mathbb Z$,
\[\kappa \colon \hh{p+1,*}{\modulesfp{\C C },\widehat{\ext}_{\C C}^{q,*}}
\To
\hh{0,*}{\modulesfp{\C C },\widehat{\ext}_{\C C}^{p+q,*}}\]
as follows. Given a cohomology class $\{\psi\}$ in the source represented by the cocycle $\psi$ and a finitely presented right $\C C$-module $M$, we consider the $p$-extension in $\modulesfp{\C C}$
\[M\stackrel{f_0}\hookrightarrow P_0\stackrel{f_1}\longrightarrow \cdots\stackrel{f_{p-1}}\longrightarrow P_{p-1}\stackrel{f_p}\twoheadrightarrow S^{p}M\]
with middle projective-injective terms obtained as the Yoneda product of the extensions defining $S^iM$, $1\leq i\leq p$, we evaluate the cocycle $\psi$ on it
\[\psi(f_p,\cdots, f_0)\in \widehat{\ext}_{\C C}^{q,*}(M,S^pM)\cong
\widehat{\ext}_{\C C}^{p+q,*}(M,M)\]
and define $\kappa(\{\psi\})(M)$ as the corresponding element on the right of the isomorphism. This isomorphism is also induced by the short exact sequences with projective-middle terms defining $S^iM$, $1\leq i\leq p$. It is easy to check that $\kappa(\{\psi\})$ does not depend on the choice of representative $\psi$, since $f_{i+1}f_i=0$ and all the $P_i$ are projective-injective. This must also be used to check that $\kappa(\{\psi\})$ is indeed a $0$-cocycle.
\end{definition}
\begin{proposition}\label{Heller_criterion_by_differential}
If $\C T$ is a small ungraded additive category over a field $k$ such that $\modulesfp{\C T}$ is Frobenius abelian and $\Sigma\colon \C T\rightarrow\C T$ is an automorphism, then the set of Puppe triangulated structures on $\C T$ with suspension functor $\Sigma$ is in bijection with the units of the bigraded ring $\hh{0,*}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$ lying in the kernel of the composite
\begin{center}
\begin{tikzcd}
\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}
\arrow[d,"d_2"]\\
\hh{2,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{2,*}}
\arrow[d,"\kappa"]\\
\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}
\end{tikzcd}
\end{center}
where $d_2$ is a second differential in the spectral sequence of Proposition \ref{spectral_sequence}.
\end{proposition}
\begin{proof}
Using \eqref{anticommutativity_cohomological}, Proposition \ref{Heller_ungraded}, \eqref{exact_cohomology_sequence}, and Corollary \ref{Heller_graded} and its proof, we see that it suffices to prove that the following diagram commutes up to sign,
\begin{center}
\begin{tikzcd}[column sep=50]
\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}
\arrow[r,"\kappa d_2"] \arrow[d, "i^*", hook]
&\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}\arrow[d, "i^*", hook]
\\
\hh{0,-1}{\modulesfp{\C T},\ext_{\C T_{\Sigma}}^{3,*}}\arrow[d, "\cong"]
&\hh{0,-1}{\modulesfp{\C T},\ext_{\C T_{\Sigma}}^{3,*}}\arrow[d, "\cong"]\\
\hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}\arrow[r, "\operatorname{id}-\tau_*^{-1}S^*"]&\hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}
\end{tikzcd}
\end{center}
Let $\varphi\in \hh{0,-1}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{3,*}}$. Our spectral sequence is the spectral sequence of a bicomplex, hence it is straightforward, although a little bit tedious, to compute $d_2(\varphi)$. First of all, given a finitely presented right $\C T_\Sigma$-module $M$ we must represent $\varphi(M)\in \ext_{\C T_{\Sigma}}^{3,-1}(M,M)$ by a right $\C T_\Sigma$-module morphism of degree $-1$
\[\tilde\varphi(M)\colon M\otimes_{\C T_{\Sigma}}B_3(\C T_{\Sigma})\To M,\]
which, by the Yoneda lemma, is the same as a collection of degree $-1$ morphisms of graded $k$-modules
\[\tilde\varphi(M)\colon M(X_0)\otimes\C T_{\Sigma}(X_1,X_0)
\otimes\C T_{\Sigma}(X_2,X_1)
\otimes\C T_{\Sigma}(X_3,X_2)
\To
M(X_3)
\]
indexed by objects $X_0,\dots, X_3$ in $\C T$. We can suppose without loss of generality that $\tilde\varphi(M)=0$ for $M$ projective-injective.
If we have a projective-injective resolution of $M$
\begin{center}
\begin{tikzcd}[column sep=5mm]
\cdots\arrow[r]&
\C T_{\Sigma}(-,X_3)\arrow[r,"d_3"]&
\C T_{\Sigma}(-,X_2)\arrow[r,"d_2"]&
\C T_{\Sigma}(-,X_1)\arrow[r,"d_1"]&
\C T_{\Sigma}(-,X_0)\ar[r, two heads,"d_0"]&
M,
\end{tikzcd}
\end{center}
there is a morphism of resolutions $\C T_{\Sigma}(-,X_*)\rightarrow M\otimes_{\C T_{\Sigma}}B_*(\C T_\Sigma)$ defined degreewise by the elements
\[
(-1)^{\frac{n(n+1)}{2}}d_0\otimes\cdots \otimes d_n\otimes\operatorname{id}_{X_n}
\]
in
\[
M(X_0)\otimes\C T_{\Sigma}(X_1,X_0)\otimes\cdots\otimes\C T_{\Sigma}(X_n,X_{n-1})\otimes\C T_{\Sigma}(X_n,X_n).
\]
Hence $\varphi(M)$ is also represented by the
morphism $\C T_{\Sigma}(-,X_3)\rightarrow M$ defined by
$\tilde{\varphi}(M)(d_0\otimes d_1\otimes d_2\otimes d_3)$.
Since $\varphi$ is a Hochschild cocycle, for each $g\colon M\rightarrow N$ in $\modulesfp{\C T_{\Sigma}}$ we must have
\[\psi(g)\colon M\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\To N,\]
of degree $-1$, i.e.
\[\psi(g)\colon M(X_0)\otimes\C T_{\Sigma}(X_1,X_0)
\otimes\C T_{\Sigma}(X_2,X_1)
\To
N(X_2)
\]
as above such that
\begin{multline}\label{equation_phi_psi}
(-1)^{\abs{g}}g\cdot \tilde{\varphi}(M)(x\otimes f_1\otimes f_2\otimes f_3)
-\tilde{\varphi}(N)(g\cdot x\otimes f_1\otimes f_2\otimes f_3)={}
\\
\psi(g)(x\cdot f_1\otimes f_2\otimes f_3)
-\psi(g)(x\otimes f_1\cdot f_2\otimes f_3)\\
+\psi(g)(x\otimes f_1\otimes f_2\cdot f_3)
-\psi(g)(x\otimes f_1\otimes f_2)\cdot f_3.
\end{multline}
Moreover, $\psi(g)$ must be $k$-linear in $g$.
The cohomology class $d_2(\varphi)$ is represented by the $2$-cocycle $\xi$ such that,
given two composable morphisms in $\modulesfp{\C T}$
\[M_2\stackrel{g_2}{\To}M_1\stackrel{g_1}{\To}M_0,\]
the element $\xi(g_1\otimes g_2)\in \ext_{\C T_{\Sigma}}^{2,\abs{g_1}+\abs{g_2}-1}(M_2,M_0)$ is represented by the map
\[(-1)^{\abs{g_1}}g_1\psi(g_2)-\psi(g_1\cdot g_2)+\psi(g_1)(g_2\otimes\operatorname{id})\colon M_2\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\To M_0.\]
According to the definition of $\kappa$, we can decompose it in two steps, $\kappa=\kappa_2\kappa_1$,
\begin{align*}
\hh{2,-1}{\modulesfp{\C T_{\Sigma} },\widehat{\ext}_{\C T_{\Sigma}}^{2,*}}
&\stackrel{\kappa_1}\To
\hh{0,-1}{\modulesfp{\C T_{\Sigma} },\widehat{\ext}_{\C T_{\Sigma}}^{2,*}(-,S)}\\
&\mathop{\To}\limits_{\kappa_2}^{\cong}
\hh{0,-1}{\modulesfp{\C T_{\Sigma} },\widehat{\ext}_{\C T_{\Sigma}}^{3,*}}.
\end{align*}
The second one, $\kappa_2$, is an isomorphism.
If we consider the short exact sequences defining $S$,
\[M\stackrel{j_M}\hookrightarrow\C T_{\Sigma}(-,X_M)\stackrel{p_M}\twoheadrightarrow S M,\]
consisting of degree $0$ morphisms,
then $\kappa_1d_2(\varphi)(M)\in \widehat{\ext}_{\C T_{\Sigma}}^{2,-1}(M,SM)$ is represented by
\[p_M\psi(j_M)+\psi(p_M)(j_M\otimes\operatorname{id})\colon M\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\To SM.\]
With the small resolution $\C T_{\Sigma}(-,X_*)$ of $M$, $\kappa_1d_2(\varphi)(M)$ is represented by
\begin{equation}\label{k_d2_phi}
-p_M \psi(j_M)(d_0\otimes d_1\otimes d_2)-\psi(p_M) (j_M\cdot d_0\otimes d_1\otimes d_2)\in SM(X_2).
\end{equation}
Using \eqref{equation_phi_psi}, it is straightfoward to check that both summands represent elements in $\widehat{\ext}_{\C T_{\Sigma}}^{2,-1}(M,SM)$, i.e.~each of them vanishes when multiplying by $d_3$ on the right.
The natural isomorphism
\begin{equation}\label{iso_S}
\widehat{\ext}_{\C T_{\Sigma}}^{2,-1}(M,SN)\cong
\widehat{\ext}_{\C T_{\Sigma}}^{3,-1}(M,N)
\end{equation}
defining $\kappa_2$ is given as follows. We pick a representative $\zeta\colon\C T_\Sigma(-,X_2)\rightarrow SN$ of an element in the source, we lift $\zeta$ along $p_N$, $\zeta=p_N\zeta'$, $\zeta'd_3$ factors (uniquely) through the inclusion $j_N$, $\zeta'd_3=j_N\zeta''$, and $\zeta''\colon\C T_{\Sigma}(-,X_3)\rightarrow N$ represents the image of the isomorphism \eqref{iso_S}. Note that, if we plug the extension defining $SM$ to the previous resolution of $M$, we obtain a projective-injective resolution of $SM$,
\begin{center}
\begin{tikzcd}[column sep=5mm]
\cdots\arrow[r]&
\C T_{\Sigma}(-,X_2)\arrow[r,"d_2"]&
\C T_{\Sigma}(-,X_1)\arrow[r,"d_1"]&
\C T_{\Sigma}(-,X_0)\arrow[r,"j_Md_0"]&
\C T_{\Sigma}(-,X_M)
\ar[r, two heads,"p_M"]&
SM,
\end{tikzcd}
\end{center}
yielding an immediate identification $\widehat{\ext}_{\C T_{\Sigma}}^{2,-1}(M,SN)=\widehat{\ext}_{\C T_{\Sigma}}^{3,-1}(SM,SN)$. Moreover, using this identification, the isomorphism \eqref{iso_S} is $S^{-1}$. Hence, if we apply it to $\varphi(SM)$ we obtain $S^{-1}\varphi(SM)=-\tau^{-1}_*S^*(\varphi)(M)$.
The first summand in \eqref{k_d2_phi} is already lifted along $p_M$. Moreover, by \eqref{equation_phi_psi},
\begin{equation*}
\psi(j_M)(d_0\otimes d_1\otimes d_2)\cdot d_3=j_M\cdot\tilde{\varphi}(M)(d_0\otimes d_1\otimes d_2\otimes d_3),
\end{equation*}
and $\tilde{\varphi}(M)(d_0\otimes d_1\otimes d_2\otimes d_3)$ represents $\varphi(M)$.
Furthermore, again by \eqref{equation_phi_psi},
\begin{equation*}
\begin{split}
\psi(p_M) (j_M\cdot d_0\otimes d_1\otimes d_2)=&-\tilde{\varphi}(SM)(p_M\otimes j_M\cdot d_0\otimes d_1\otimes d_2)\\&
+\psi(p_M) (\operatorname{id}_{X_M}\otimes j_M\cdot d_0\otimes d_1)\cdot d_2,
\end{split}
\end{equation*}
and $\tilde{\varphi}(SM)(p_M\otimes j_M\cdot d_0\otimes d_1\otimes d_2)$ represents $\varphi(SM)$.
Hence, by the previous paragraph, applying $\kappa_2$ to the element represented by the second summand in \eqref{k_d2_phi} we obtain $\tau^{-1}_*S^*(\varphi)(M)$.
Summing up, we have checked that $\kappa d_2(\varphi)(M)=-\varphi(M)+\tau^{-1}_*S^*(\varphi)(M)$.
This finishes the proof.
\end{proof}
\section{The octahedral axiom}\label{octahedral_axiom_section}
Verdier's octahedral axiom, unlike the rest of axioms for a triangulated category, does not seem to have an algebraic characterization. Nevertheless we have the following algebraic sufficient condition. After Proposition \ref{Heller_criterion_by_differential}, this sufficient condition can be regarded as a strengthening of Heller's anticommutativity condition. A rather close strengthening indeed. It actually reflects in a very precise way the known fact that the octahedral axiom is at the bottom of the coherence hierarchy of enhancements. The proof is rather lengthy because we must prove the octahedral axiom in a non-standard way, but we obtain as a corollary an interesting characterization of pre-triangulated DG- or $A$-infinity categories over a field, in the sense of Bondal and Kapranov.
\begin{theorem}\label{octahedral_theorem}
If $\C T$ is a small ungraded idempotent complete additive category over a field $k$ such that $\modulesfp{\C T}$ is Frobenius abelian, $\Sigma\colon \C T\rightarrow\C T$ is an automorphism, and $\varphi\in \hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}$ corresponds to a Puppe triangulated structure on $\C T$ with suspension $\Sigma$ such that $d_2(\varphi)=0$, then this Puppe triangulated structure satisfies the octahedral axiom.
\end{theorem}
\begin{proof}
Neeman's mapping cone criterion \cite[Definition 1.3.13, Proposition 1.4.6, and Remark 1.4.7]{neeman_triangulated_2001} seems to be the preferred route to prove the octahedral axiom for triangulated categories with no (known) models \cite{amiot_structure_2007, muro_triangulated_2007}.
We must show that any map between the bases of two exact triangles can be extended to a triangle morphisms whose mapping cone is exact, i.e.~for any commutative diagram of solid arrows between exact triangles
\begin{center}
\begin{tikzcd}
X\arrow[r,"f"]\arrow[d," h_1"]&Y\arrow[r,"i"]\arrow[d,"h_2"]& C_f\arrow[r,"q"]\arrow[d,"h_3",dashed]& \Sigma X\arrow[d,"\Sigma h_1"]\\
X'\arrow[r,"f'"]&
Y'\arrow[r,"i'"]&
C_{f'}\arrow[r,"q'"]
&\Sigma X'
\end{tikzcd}
\end{center}
we can find a filler $h_3$ whose mapping cone
\begin{center}
\begin{tikzcd}[column sep=16mm, ampersand replacement=\&]
Y\oplus X'\arrow[r, "
{\left(
\begin{smallmatrix}
-i&0\\
h_2&f'
\end{smallmatrix}
\right)}
"]\&C_f\oplus Y'\arrow[r,"{\left(\begin{smallmatrix}
-q&0\\h_3&i'
\end{smallmatrix}\right)}"]\& \Sigma X\oplus C_{f'}
\arrow[r,"{\left(\begin{smallmatrix}
-\Sigma f&0\\h_3&q'
\end{smallmatrix}\right)}"]\& \Sigma Y\oplus \Sigma X'
\end{tikzcd}
\end{center}
is an exact triangle.
We claim that, if this holds for $(h_1,h_2)$ then it also holds for $(h_1',h_2')$ provided there are maps
\[X\stackrel{k_1}\longrightarrow T\stackrel{k_2}{\longrightarrow} X',\qquad \Theta\colon Y\longrightarrow X',\]
such that
\begin{align*}
f'k_2&=0,&
h_1'-h_1-k_2k_1&=\Theta f.
\end{align*}
We check this claim by playing with Puppe's axioms. Indeed, by the first equation $k_2$ must factor through $\Sigma^{-1}q'$, so we can rephrase our conditions as follows: there exist maps
\[\Sigma^{-1}\Psi\colon X\longrightarrow \Sigma^{-1}C_{f'},\qquad \Theta\colon Y\longrightarrow X',\]
such that
\begin{align*}
h_1'-h_1&=\Theta f+\Sigma^{-1}(q'\Psi).
\end{align*}
We have
\begin{equation*}
\begin{split}
(h_2'-h_2-f'\Theta)f&=(h_2'-h_2)f-f'\Theta f\\
&=f'(h_1'-h_1)-f'(\Theta f)\\
&=f'(h_1'-h_1-\Theta f)\\
&=\underbrace{f'(\Sigma^{-1}q')}_{=0}(\Sigma^{-1}\Psi)=0.
\end{split}
\end{equation*}
Therefore, there exists $\Phi\colon C_f\rightarrow Y'$ such that \[h_2'-h_2=\Phi i+f'\Theta.\]
We claim that, if we define \[h_3'=h_3+i'\Phi+\Psi q\] then $(h_1',h_2',h_3')$ is a morphism of triangles. We only have to check that the two squares on the right, those containing $h_3'$, commute. This amounts to
\begin{align*}
h_3'i&=(h_3+i'\Phi+\Psi q)i&
q'h_3'&=q'(h_3+i'\Phi+\Psi q)
\\
&=h_3i+i'\Phi i+\Psi qi&
&=q'h_3+q'i'\Phi+q'\Psi q\\
&=i'h_2+i'\Phi i&
&=(\Sigma h_1)q+q'\Psi q\\
&=i'h_2+i'\Phi i+i'f'\Theta\quad {\scriptstyle (i'f'=0)}&
&=(\Sigma h_1)q+q'\Psi q+\Sigma(\Theta f)q\quad {\scriptstyle ((\Sigma f)q=0)}\\
&=i'(h_2+\Phi i+f'\Theta)&
&=(\Sigma h_1+q'\Psi +\Sigma(\Theta f))q\\
&=i'h_2',&
&=(\Sigma h_1')q.\\
\end{align*}
The map of triangles $(h_1',h_2',h_3')$ is homotopic to $(h_1,h_2,h_3)$ in the sense of \cite[Definition 1.3.2]{neeman_triangulated_2001} by construction. This implies that the mapping cone of the former is isomorphic to the mapping cone of the later \cite[Lemma 1.3.3]{neeman_triangulated_2001}, hence it is also exact.
Exact triangles can be made into a category whose maps are just morphisms between the bases, i.e.~pairs $(h_1,h_2)$ as above. We can even form the quotient category under the equivalence relation $(h_1,h_2)\sim(h_1',h_2')$ defined by the existence of $\Sigma^{-1}\Psi$ and $\Theta$ as above. The quotient category is equivalent to $\modulesst{\C T}$, and the equivalence is realized by the functor sending an exact triangle $(f,i,q)$ to $\ker \C T(-,f)$ (this kernel is taken in $\modulesfp{\C T}$). This can be easily checked by using the alternative descriptions of the abelianization $\modulesfp{\C T}$ in \cite[5.1]{neeman_triangulated_2001} and the construction of the quotient category $\modulesst{\C T}$. Hence, we have just proved that the property we have to check only depends on the morphism $\ker \C T(-,f)\rightarrow \ker \C T(-,f')$ in $\modulesst{\C T}$ induced by $(h_1,h_2)$. It does not even depend on the triangles $(f,i,q)$ and $(f',i',q')$ lifting $\ker \C T(-,f)$ and $\ker \C T(-,f')$ through the previous equivalence of categories.
Let us denote $A=\ker \C T(-,f)$ and $A'=\ker \C T(-,f')$. Using the suspension functor $S$ in the stable module category we get
\[\homst_{\C T}(A,A')\cong \ext_{\C T}^{1}(S A,A').\] We can assume that $SA$ is defined by the short exact sequence
\[A\hookrightarrow\C T(-,X)\twoheadrightarrow SA\]
in $\modulesfp{\C T}$ arising from the exact triangle $(f,i,q)$, hence $SA$ is also $\ker \C T(-,i)$. The short exact sequence
\[A'\hookrightarrow B\twoheadrightarrow SA\]
representing the extension corresponding to the map $A\rightarrow A'$ induced by $(h_1,h_2)$ can be obtained from the following diagram
\begin{center}
\begin{tikzcd}[column sep=16mm, ampersand replacement=\&]
X'\arrow[r,"f'"]\arrow[d, hook]\&
Y' \arrow[d, hook]
\\
Y\oplus X'\arrow[d, two heads]\arrow[r, "
{\left(
\begin{smallmatrix}
-i&0\\
h_2&f'
\end{smallmatrix}
\right)}
"]\&C_f\oplus Y'\arrow[d, two heads]\\
Y\arrow[r,"-i"] \& C_f
\end{tikzcd}
\end{center}
first embedding it in $\modulesfp{\C T}$ through the Yoneda inclusion and then taking kernels of horizontal arrows. Here, the vertical arrows are the obvious inclusions and projections of factors of a direct sum. Indeed, by the following paragraph we obtain a short exact sequence when taking kernels.
For any choice of filler $h_3$ in the very first diagram of this proof, the mapping cone fits in the following commutative diagram of triangles
\begin{center}
\begin{tikzcd}[column sep=16mm, ampersand replacement=\&]
X'\arrow[r,"f'"]\arrow[d, hook]\&
Y'\arrow[r,"f'"] \arrow[d, hook]
\&C_{f'}\arrow[d, hook]\arrow[r,"q'"]\&\Sigma X'\arrow[d, hook]
\\
Y\oplus X'\arrow[d, two heads]\arrow[r, "
{\left(
\begin{smallmatrix}
-i&0\\
h_2&f'
\end{smallmatrix}
\right)}
"]\&C_f\oplus Y'\arrow[d, two heads]\arrow[r,"{\left(\begin{smallmatrix}
-q&0\\h_3&i'
\end{smallmatrix}\right)}"]\& \Sigma X\oplus C_{f'}\arrow[d, two heads]
\arrow[r,"{\left(\begin{smallmatrix}
-\Sigma f&0\\\Sigma h_1&q'
\end{smallmatrix}\right)}"]\& \Sigma Y\oplus \Sigma X'\arrow[d, two heads]\\
Y\arrow[r,"-i"] \& C_f\arrow[r,"-q"] \& \Sigma X \arrow[r,"-\Sigma f"]\&\Sigma Y
\end{tikzcd}
\end{center}
with exact top and bottom triangles. In particular, the top and bottom sequences can be extended to the right in the usual way, $3$-periodic twisted by $\Sigma$. These sequences become injective resolutions of $A$ and $SA'$ in $\modulesfp{\C T}$ via the Yoneda inclusion. By the long exact homology sequence, we can actually do the same with the middle sequence, despite it need not be an exact triangle. This yields a resolution of $B$.
Now, we consider the following induced diagram in $\modulesfp{\C T}$ where, abusing notation, we identify each object in $\C T$ with its Yoneda image in $\modulesfp{\C T}$,
\begin{center}
\begin{tikzcd}[column sep=13mm, ampersand replacement=\&]
A'\arrow[r, hook]\arrow[d, hook]\&X'\arrow[r,"f'"]\arrow[d, hook]\&
Y'\arrow[r,"f'"] \arrow[d, hook]
\&C_{f'}\arrow[d, hook]\arrow[r, two heads]\&\Sigma A'\arrow[d, hook]
\\
B\arrow[r, hook]\arrow[d, two heads]\&Y\oplus X'\arrow[d, two heads]\arrow[r, "
{\left(
\begin{smallmatrix}
-i&0\\
h_2&f'
\end{smallmatrix}
\right)}
"]\&C_f\oplus Y'\arrow[d, two heads]\arrow[r,"{\left(\begin{smallmatrix}
-q&0\\h_3&i'
\end{smallmatrix}\right)}"]\& \Sigma X\oplus C_{f'}\arrow[d, two heads]
\arrow[r, two heads]\& \Sigma B\arrow[d, two heads]\\
SA\arrow[r, hook]\&Y\arrow[r,"-i"] \& C_f\arrow[r,"-q"] \& \Sigma X \arrow[r, two heads]\& \Sigma SA
\end{tikzcd}
\end{center}
Here, the top and bottom extensions represent $\varphi(\Sigma A')$ and $\varphi(\Sigma SA)$, respectively, since they have been obtained from exact triangles, see Section \ref{heller_section}. We have to show that we can find $h_3$ such that the middle extension represents $\varphi(\Sigma B)$. Since we have checked that we can suitably modify $h_1$ and $h_2$, it suffices to show that we can find a representative of $\varphi(\Sigma B)$
\[B\hookrightarrow P_2\rightarrow P_1\rightarrow P_0\rightarrow \Sigma B\] fitting into a commutative diagram with exact columns,
\begin{center}
\begin{tikzcd}[column sep=13mm, ampersand replacement=\&]
A'\arrow[r, hook]\arrow[d, hook]\&X'\arrow[r,"f'"]\arrow[d, hook]\&
Y'\arrow[r,"f'"] \arrow[d, hook]
\&C_{f'}\arrow[d, hook]\arrow[r, two heads]\&\Sigma A'\arrow[d, hook]
\\
B\arrow[r, hook]\arrow[d, two heads]\&P_2\arrow[d, two heads]\arrow[r]\&P_1\arrow[d, two heads]\arrow[r]\& P_0\arrow[d, two heads]
\arrow[r, two heads]\& \Sigma B\arrow[d, two heads]\\
SA\arrow[r, hook]\&Y\arrow[r,"-i"] \& C_f\arrow[r,"-q"] \& \Sigma X \arrow[r, two heads]\& \Sigma SA
\end{tikzcd}
\end{center}
since then the middle extension must come from a triangle as above. We can even replace the given representatives of $\varphi(\Sigma A')$ and $\varphi(\Sigma SA)$ with any others. Indeed, using standard arguments from homological algebra, like uniqueness of resolutions, it is easy to check that if we can find this diagram for two given representatives then we can also do it for any others. It is not even important that the middle $\C T$-modules are finitely presented. Note also that any short exact sequence in $\modulesfp{\C T}$ can arise as \[A'\hookrightarrow B\twoheadrightarrow SA.\]
Hence, below we consider an arbitrary one.
In order to simplify notation, we move from $\modulesfp{\C T}$ to $\modulesfp{\C T_\Sigma}$ in the rest of this proof (it is equivalent since the former abelian category is the degree $0$ part of the latter graded abelian category). We now recall some notation and facts from the proof of Proposition \ref{Heller_criterion_by_differential} that we also need here.
Let $M$ be a finitely presented $\C T_\Sigma$-module. The element $\varphi(M)\in \ext_{\C T_{\Sigma}}^{3,-1}(M,M)$ is represented by a morphism \[\tilde\varphi(M)\colon M\otimes_{\C T_{\Sigma}}B_3(\C T_{\Sigma})\To M.\] A representing extension can be obtained by taking push-out along $\tilde\varphi(M)$, as in the following diagram
\begin{center}
\begin{tikzcd}[column sep=3.5mm]
M\otimes_{\C T_{\Sigma}}B_3(\C T_{\Sigma})\arrow[r,"d"]\arrow[d,"\tilde\varphi(M)"'] \arrow[dr, phantom, "\text{\scriptsize push}"]&
M\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\arrow[r,"d"]\arrow[d]&
M\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r,"d"]\arrow[d, equal]&
M\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r,"\epsilon"]\arrow[d, equal]&
M\arrow[d, equal]\\
M\arrow[r, hook]&
P_M\arrow[r]&
M\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]&
M\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]&
M
\end{tikzcd}
\end{center}
Here, in the top row, $d$ actually means $\operatorname{id}_M\otimes d$, where $d$ is the bar complex differential, and similarly for the augmentation $\epsilon$.
Given a morphism $g\colon M\rightarrow N$ in $\modulesfp{\C T_\Sigma}$, the cocycle condition \[(-1)^{\abs{g}}g\cdot \varphi(M)=\varphi(N)\cdot g\]
is realized by the existence of
\[\psi(g)\colon M\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\To N,\]
which is $k$-linear in $g$, such that
\[\psi(g)(d\otimes \operatorname{id})=(-1)^{\abs{g}}g\tilde \varphi(M)-\tilde\varphi(N) (g\otimes\operatorname{id}).\]
This allows the definition of a morphism of extensions
\begin{center}
\begin{tikzcd}
M\arrow[r, hook]\arrow[d,"(-1)^{\abs{g}}g"']&
P_M\arrow[r]\arrow[d,"P_g"]&
M\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]\arrow[d,"g\otimes\operatorname{id}"]&
M\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]\arrow[d,"g\otimes\operatorname{id}"]&
M\arrow[d,"g"]\\
N\arrow[r, hook]&
P_N\arrow[r]&
N\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]&
N\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]&
N
\end{tikzcd}
\end{center}
where the morphism $P_g$ is defined by applying the universal property of a push-out to the following diagram
\begin{center}
\begin{tikzcd}
M\otimes_{\C T_{\Sigma}}B_3(\C T_{\Sigma})\arrow[r,"\operatorname{id}_M\otimes d"]\arrow[d,"\tilde\varphi(M)"] \arrow[dr, phantom, "\text{\scriptsize push}"]\arrow[d]&
M\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\arrow[d]\arrow[ddr, bend left=10, "\parbox{18mm}{\scriptsize sum of the two dashed paths}", near end]\arrow[r,"g\otimes\operatorname{id}", dashed]\arrow[ldd, dashed, bend left=10, "\psi(g)"]&
N\otimes_{\C T_{\Sigma}}B_2(\C T_{\Sigma})\arrow[dd, bend left=75, dashed]\\
M\arrow[r, hook, crossing over]\arrow[d,"(-1)^{\abs{g}}g"']&P_M\arrow[dr, dotted, "P_f"]&\\
N\arrow[rr, hook]&
&
P_N
\end{tikzcd}
\end{center}
If $d_2(\varphi)=0$ then, for any pair of composable morphisms in $\modulesfp{\C T_\Sigma}$,
\[L\stackrel{f}{\To}M\stackrel{g}{\To }N,\]
there exists a morphism
\[\zeta(g\otimes f)\colon L\otimes B_1(\C T_\Sigma)\longrightarrow N,\]
bilinear in $f$ and $g$, such that
\[\zeta(g\otimes f)(\operatorname{id}\otimes d)=(-1)^{\abs{g}}g\psi(f)-\psi(gf)+\psi(g)(f\otimes\operatorname{id}).\]
In particular, $P_gP_f-P_{gf}$ is the composite
\[P_L\longrightarrow L\otimes B_1(\C T_\Sigma)\stackrel{\zeta(g\otimes f)}\longrightarrow N\hookrightarrow P_N.\]
Assume $g\colon M\twoheadrightarrow N$ is surjective. Then we can find a lift \[\zeta'\colon L\otimes B_1(\C T_\Sigma)\longrightarrow M\] of $\zeta(g\otimes f)$ along $g$, and if we define $\delta$ as
\[(-1)^{\abs{g}+1}\delta\colon P_L\longrightarrow L\otimes B_1(\C T_\Sigma)\stackrel{\zeta'}\longrightarrow M\hookrightarrow P_M,\]
then
\[P_g(P_f+\delta)=P_{gf}.\]
Moreover,
\begin{center}
\begin{tikzcd}
L\arrow[r, hook]\arrow[d,"(-1)^{\abs{f}}f"']&
P_L\arrow[r]\arrow[d,"P_f+\delta"]&
L\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]\arrow[d,"f\otimes\operatorname{id}"]&
L\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]\arrow[d,"f\otimes\operatorname{id}"]&
L\arrow[d,"f"]\\
M\arrow[r, hook]&
P_M\arrow[r]&
M\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]&
M\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]&
M
\end{tikzcd}
\end{center}
is still a map of extensions.
We claim that, if
\[L\stackrel{f}{\hookrightarrow}M\stackrel{g}{\twoheadrightarrow}N\]
is a short exact sequence, then so are the colums of
\begin{center}
\begin{tikzcd}
L\arrow[r, hook]\arrow[d, hook,"(-1)^{\abs{f}}f"']&
P_L\arrow[r]\arrow[d,"P_f+\delta"]&
L\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]\arrow[d, hook,"f\otimes\operatorname{id}"]&
L\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]\arrow[d, hook,"f\otimes\operatorname{id}"]&
L\arrow[d, hook,"f"]\\
M\arrow[r, hook]\arrow[d, two heads,"(-1)^{\abs{g}}g"']&
P_M\arrow[r]\arrow[d,"P_g"]&
M\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]\arrow[d, two heads,"g\otimes\operatorname{id}"]&
M\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]\arrow[d, two heads,"g\otimes\operatorname{id}"]&
M\arrow[d, two heads,"g"]\\
N\arrow[r, hook]&
P_N\arrow[r]&
N\otimes_{\C T_{\Sigma}}B_1(\C T_{\Sigma})\arrow[r]&
N\otimes_{\C T_{\Sigma}}B_0(\C T_{\Sigma})\arrow[r]&
N
\end{tikzcd}
\end{center}
The only column where the claim is not obvious is the second one. We know at least that $P_g(P_f+\delta)=P_{gf}=P_0=0$. If we write $Q_L,Q_M,Q_N$ for the kernels of the horizontal arrows between the third and fourth columns, the long exact homology sequence yields a short exact sequence
\[Q_L\hookrightarrow Q_M\twoheadrightarrow Q_N.\]
These are also the images of the horizontal arrows between the second and third columns. Hence the previous short exact sequence fits in a commutative diagram
\begin{center}
\begin{tikzcd}
L\arrow[r, hook]\arrow[d, hook,"(-1)^{\abs{f}}f"']&
P_L\arrow[r, two heads]\arrow[d,"P_f+\delta"]&
Q_L\arrow[d, hook]\\
M\arrow[r, hook]\arrow[d, two heads,"(-1)^{\abs{g}}g"']&
P_M\arrow[r,two heads]\arrow[d,"P_g"]&
Q_M\arrow[d, two heads]\\
N\arrow[r, hook]&
P_N\arrow[r]&
Q_N
\end{tikzcd}
\end{center}
Applying the snake lemma to the top and bottom maps of short exact sequences we see that $P_f+\delta$ is injective and $P_g$ is surjective. Moreover, since the middle column composes to zero, the diagram and the snake lemma also yield a map of extensions
\begin{center}
\begin{tikzcd}
L\arrow[r, hook]\arrow[d, equal]&
P_L\arrow[r, two heads]\arrow[d]&
Q_L\arrow[d, equal]\\
L\arrow[r, hook]&
\ker P_g\arrow[r]&
Q_L
\end{tikzcd}
\end{center}
which concludes the proof of exactness since the middle vertical arrow is necessarily an isomorphism by the five lemma.
\end{proof}
Following \cite{bondal_enhanced_1991}, we say that a DG- or $A$-infinity category $\C C$ is \emph{pre-triangulated} if $H^*(\C C)$ is weakly stable, i.e.~up to equivalence it is of the form $\C T_\Sigma$, and the usual Massey product in the cohomology $H^*(\C C)\simeq\C T_{\Sigma}$ induces a triangulated structure on $H^0(\C C)\simeq\C T$ with suspension $\Sigma$. Over a field, the \emph{universal Massey product} of a DG- or $A$-infinity category $\C C$,
\[\{m_3\}\in \hh{3,-1}{H^*(\C C),H^*(\C C)},\]
is the universal Massey product of any minimal model (it does not depend on the choice).
The following result is a direct consequence of Proposition \ref{Heller_criterion_by_differential} and Theorem \ref{octahedral_theorem}.
\begin{corollary}\label{pretriangulated}
Let $\C C$ be a DG- or $A$-infinity category over a field $k$ such that idempotents in $H^0(\C C)$ split. The following statements are equivalent:
\begin{itemize}
\item $\C C$ is pre-triangulated.
\item $H^*(\C C)$ is weakly stable, $\modulesfp{H^0(\C C)}$ is Frobenius abelian, and the image of the universal Massey product $\{m_3\}\in \hh{3,-1}{H^*(\C C), H^*(\C C)}$ along the edge morphism \[\hh{3,-1}{H^*(\C C), H^*(\C C)}\longrightarrow \hh{0,-1}{\modulesfp{H^*(\C C)},\ext_{H^*(\C C)}^{3,\ast}}\] in Theorem \ref{theorem_edge_morphism} is a unit in the bigraded algebra \[\hh{0,*}{\modulesfp{H^*(\C C)},\widehat{\ext}_{H^*(\C C)}^{\star,\ast}}.\]
\end{itemize}
\end{corollary}
\section{The topological case}\label{topological_section}
In this section we move to a non-additive setting. Let $\operatorname{Set}$ be the category of \emph{graded sets} $X=\{X_n\}_{n\in\mathbb Z}$. We endow it with the closed symmetric monoidal structure defined as \[(X\boxtimes Y)_n=\coprod_{n=p+q}X_p\times Y_q.\] In this section, a \emph{graded category} is a category enriched in $\operatorname{Set}$. We will also use graded liner categories, always defined over $\mathbb Z$. Here we will specify when a given (graded) category is linear.
The obvious forgetful functor
\[\modules{\mathbb Z}\longrightarrow\operatorname{Set}\]
is lax monoidal. Given two graded abelian groups $A$ and $B$, the natural map
\[A\boxtimes B\longrightarrow A\otimes B\]
is given by the universal bilinear maps $A_p\times B_q\rightarrow A_p\otimes B_q$.
The forgetful functor has a left adjoint, the \emph{free graded abelian group} functor
\[\operatorname{Set}\longrightarrow\modules{\mathbb Z}\colon X\mapsto\mathbb Z(X),\]
which is strict monoidal. Neither is symmetric because there is no way to encode the Koszul sign rule in $\operatorname{Set}$. Nevertheless, that is sufficient to functorially define the graded linear category $\mathbb Z\C C$ associated to a graded category $\C C$, that we call \emph{linearization}. It has the same objects as $\C C$ and morphism objects \[(\mathbb Z\C C)(X,Y)=\mathbb Z\C C(X,Y),\]
and composition is defined by
\[\mathbb Z\C C(Y,Z)\otimes \mathbb Z\C C(X,Y)\cong \mathbb Z(\C C(Y,Z)\boxtimes \C C(X,Y))\longrightarrow \mathbb Z\C C(X,Z).\]
Linearization is the left adjoint of the forgetful functor from graded linear categories to graded categories. For the sake of simplicity, the linearization of a graded functor between graded categories $F\colon \C C\rightarrow\C D$ will also be denoted by $F\colon \mathbb Z\C C\rightarrow\mathbb Z\C D$. The lack of compatibility with the symmetry constraint implies that the linearization functor does not take tensor products of graded categories to tensor products of graded linear categories, but this will be irrelevant because we will only consider the latter.
The \emph{cohomology} of a graded category $\C C$ with coefficients in a $\mathbb Z\C C$-bimodule $M$ is defined as
\[H^{\star,*}(\C C,M)=\hh{\star,*}{\mathbb Z\C C, M}.\]
A $\mathbb Z\C C$-bimodule can also be described as a family of graded abelian groups $M(X,Y)$ indexed by pairs of objects $X,Y$ in $\C C$
and, for each four objects $X,X',Y,Y'$ in $\C C$, a degree $0$ map of graded sets
\begin{align*}
\C C(Y,Y')\boxtimes M(X,Y)\boxtimes \C C(X',X)&\longrightarrow M(X',Y'),\\
(g, x, f)&\;\mapsto\; g\cdot x\cdot f,
\end{align*}
or equivalently, degree $0$ maps of graded sets, $p,n,q\in\mathbb Z$,
\begin{align*}
\C C^p(Y,Y')\times M^n(X,Y)\times \C C^q(X',X)&\longrightarrow M^{p+n+q}(X',Y'),\\
(g, x, f)&\;\mapsto\; g\cdot x\cdot f,
\end{align*}
which are linear in $x$ and satisfy the usual associativity and unit conditions.
The previous cohomology of graded categories satisfies the same functoriality properties as Hochschild cohomology, compare \cite{muro_functoriality_2006}.
The ungraded version was also considered in \cite{mitchell_rings_1972}. The cochain complex $\hc{\star, \ast}{\mathbb Z\C C,M}$ defining $H^{\star,*}(\C C,M)$ can also be described as
\[\hc{n,\ast}{\mathbb Z\C C,M}=\prod_{X_0,\dots, X_n} \hom^*_{\operatorname{Set}}(\C C(X_1,X_{0})\times\cdots\times\C C(X_n,X_{n-1}), M(X_n,X_0)),\]
where $\hom^*_{\operatorname{Set}}$ is the inner $\hom$ in graded sets. The differential is given by \eqref{hochschild_differential}.
The new cohomology theory makes sense even if $\C C$ is additive. In this case, it is (or deserves to be called) the \emph{topological Hochschild cohomology} or \emph{Mac Lane cohomology} of $\C C$, see \cite{pirashvili_mac_1992, jibladze_cohomology_1991}. Moreover, the obvious linear functor $\mathbb Z\C C\rightarrow\C C$ induces \emph{comparison morphisms}
\[\hh{\star,*}{\C C,M}\To H^{\star,*}(\C C,M)\]
for any $\C C$-bimodule $M$. This morphism is an isomorphism for $\star=0$ since the end of $M$ regarded as a $\C C$-bimodule is the same as if we regard it as a $\mathbb Z\C C$-bimodule, because $\mathbb Z\C C\rightarrow\C C$ is full. In particular, we can replace $HH^{0,*}$ with $H^{0,*}$ in all results of Sections \ref{heller_section} and \ref{toda_brackets_section}. At the level of cochains, the comparison morphism
\[\hc{\star,*}{\C C,M}\hookrightarrow \hc{\star,*}{\mathbb Z\C C,M}\]
is the inclusion of multilinear cochains.
Given a plain ungraded category $\C T$ and an automorphism $\Sigma\colon \C T\rightarrow\C T$, the graded category $\C T_\Sigma$ also makes sense in the non-additive context, and it is compatible with linearization, $(\mathbb Z\C T)_{\Sigma}=\mathbb Z(\C T_\Sigma)$. In particular, if $M$ is an ungraded $\mathbb Z\C T$-bimodule equipped with an isomorphism $\tau\colon M\cong M(\Sigma,\Sigma)$
such that, given
$g\in\C T(Y,Y')$, $x\in M(X,Y)$, and $f\in\C T(X',X)$,
\[\tau(g\cdot x\cdot f)=(\Sigma g)\cdot\tau(x)\cdot(\Sigma f),\]
we can define the $\C T_\Sigma$-bimodule $M_\tau$ as in Section \ref{Hochschild_cohomology_of_categories}. Proposition \ref{graded_ungraded_long_exact_sequence} applies, so we have a long exact sequence
\begin{equation}\label{exact_cohomology_sequence_top}
\begin{tikzcd}[row sep=5mm]
\vdots\arrow[d]\\
\hbw{n,*}{\C T_{\Sigma},M_{\tau}}\arrow[d, "i^*"]\\
\hbw{n,*}{\C T,M_{\tau}}\arrow[d, "1-\tau_*^{-1}\Sigma^*"]\\
\hbw{n,*}{\C T,M_{\tau}}\arrow[d]\\
\hbw{n+1,*}{\C T_{\Sigma},M_{\tau}}\arrow[d]\\
\vdots
\end{tikzcd}
\end{equation}
This, or rather the proof of Proposition \ref{graded_ungraded_long_exact_sequence}, shows that $H^{\star,*}(\C T_{\Sigma},M_\tau)$ coincides with the translation cohomology of the pair $(\Sigma,\tau)$ defined in \cite{baues_homotopy_2007} and extensively used in \cite{baues_cohomologically_2008}.
If $\C M$ is a stable model category, its homotopy category $\C T=\operatorname{Ho}\C M$ is triangulated \cite{hovey_model_1999} with the well known suspension functor $\Sigma\colon \C T\rightarrow \C T$. Moreover, objects, maps, and tracks (i.e.~homotopy classes of homotopies relative to the boundary) define a topological analogue of the universal Massey product, that we call \emph{universal Toda bracket} \cite[Remark 5.9]{baues_homotopy_2007},
\[\langle\C M\rangle\in H^{3,-1}(\C T_\Sigma,\C T_\Sigma),\]
save for the fact that the category $\C T$ may be big, but we can replace $\C T$ with any small triangulated subcategory of $\operatorname{Ho}\C M$, whose cohomology is then well defined. Hence, any topological enhancement of a small triangulated category defines a universal Toda bracket.
A representing cocycle $m_3$ for $\langle\C M\rangle$ is formally a ternary operation as in the introduction, except for the fact that it need not be multilinear. Universal Toda brackets have been extensively studied in the ungraded setting, i.e.~the image of $\langle\C M\rangle$ along the morphism
\[H^{3,-1}(\C T_\Sigma,\C T_\Sigma)\longrightarrow H^{3,-1}(\C T,\C T_\Sigma)\cong H^3(\C T,\C T(\Sigma,-))\]
fitting in a long exact sequence as above.
They indeed determine all homotopically defined Toda brackets in $\operatorname{Ho}\C M$ \cite{baues_cohomology_1989}, which characterize its triangulated structure in the way explained in the introduction.
The spectral sequence in Proposition \ref{spectral_sequence} is also defined in the current non-additive context but the proof, although similar, needs a couple of significant modifications.
\begin{proposition}\label{spectral_sequence_top}
If $\C T$ is a small additive category such that $\modulesfp{\C T}$ is abelian and $\Sigma\colon\C T\rightarrow\C T$ is an automorphism, there is a first quadrant cohomological spectral sequence of graded abelian groups
\[E_2^{p,q}=H^{p,*}(\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{p,*})\Longrightarrow H^{p+q,*}(\C T_{\Sigma},\C T_{\Sigma}).\]
\end{proposition}
\begin{proof}
Now the spectral sequence is associated to the following bicomplex of graded modules $C^{\star,\bullet}$,
\[\hom_{\mathbb Z\modulesfp{\C T_{\Sigma}}^{\env}}^*(B_{\star}(\mathbb Z\modulesfp{\C T_{\Sigma}}),\hom_{\C T_{\Sigma}}^*(\mathbb Z(-)\otimes_{\mathbb Z\C T_{\Sigma}}B_{\bullet}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma},-)).\]
Here we use the \emph{forceful linearization} of right $\C T_\Sigma$-modules, which is a non-additive functor \[\mathbb Z(-)\colon \modules{\C T_{\Sigma}}\longrightarrow\modules{\mathbb Z\C T_{\Sigma}}\]
defined as follows. Given a right $\C T_{\Sigma}$-module $M$, $\mathbb Z(M)(X)=\mathbb Z(M(X))$ for any object $X$ in $\C T_\Sigma$ and the right action of $\mathbb Z\C T_{\Sigma}$ is given by
\[\begin{split}
\mathbb Z(M)(X)\otimes\mathbb Z\C T_{\Sigma}(X',X)&\cong\mathbb Z(M(X)\boxtimes \C T_{\Sigma}(X',X))\\&\rightarrow \mathbb Z(M(X)\otimes \C T_{\Sigma}(X',X))\\&\rightarrow \mathbb Z(M)(X').
\end{split}\]
Moreover, given a morphism of right $\C T_{\Sigma}$-modules $f\colon M\rightarrow N$ of any degree, the induced morphism $\mathbb Z(f)\colon \mathbb Z(M)\rightarrow \mathbb Z(N)$ is given by the free abelian group homomorphisms $\mathbb Z(f)(X)\colon \mathbb Z(M)(X)\rightarrow \mathbb Z(N)(X)$ defined by $f(X)\colon M(X)\rightarrow N(X)$ on the bases, where $X$ is any object in $\C T_\Sigma$.
We remark for later use that the forceful linearization functor preserves colimits, since the free grade dabelian group functor, which is a left adjoint, preserves colimits, and colimits of right modules are computed pointwise.
An element of $C^{p,q}$ is the same a family of maps of graded sets
\[\prod_{i=1}^p\hom^*_{\C T_\Sigma}(M_{i},M_{j-1})
\times M_p(X_0)\times
\prod_{j=1}^q\C T_\Sigma(X_{j},X_{j-1})\longrightarrow M_0(X_q)\]
indexed by all sequences of objects $M_0,\dots, M_p$ in $\modulesfp{\C T_\Sigma}$ and $X_0,\dots, X_q$ in $\C T_\Sigma$. With this description, the horizontal and vertical differentials are again \eqref{differentials_bicomplex}.
We must identify the $E_2$-term and the target of the spectral sequence. First, observe that for each finitely presented right $\C T_{\Sigma}$-module $M$, the complex of projective right $\C T_{\Sigma}$-modules
\[\mathbb Z (M)\otimes_{\mathbb Z\C T_{\Sigma}}B_{\bullet}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma}\]
is the standard complex computing the André--Quillen homology $H_*(M,y)$ of $M$ with coefficients in the Yoneda inclusion $y\colon \C T_{\Sigma}\hookrightarrow\modulesfp{\C T_{\Sigma}}$ in the following graded \emph{catégorie avec modèles munis de coefficients}, using André's terminology, see \cite[Chapitre I]{andre_methode_1967}, \[\modulesfp{\C T_{\Sigma}}\supset\C T_{\Sigma}\stackrel{y}{\longrightarrow}\modulesfp{\C T_{\Sigma}}.\]
Indeed,
\begin{multline*}
\mathbb Z (M)\otimes_{\mathbb Z\C T_{\Sigma}}B_{n}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma}\\
=\bigoplus_{X_0,\dots, X_n} \mathbb Z(M(X_0))\otimes\mathbb Z\C T_{\Sigma}(X_1,X_{0})\otimes\cdots\otimes\mathbb Z\C T_{\Sigma}(X_n,X_{n-1})\otimes\C T_{\Sigma}(-,X_n)\\
=\bigoplus_{M\leftarrow X_0\leftarrow \cdots\leftarrow X_n}\C T_{\Sigma}(-,X_n).
\end{multline*}
Unlike in Proposition \ref{spectral_sequence}, we here restrict the statement to graded linear categories of the form $\C T_{\Sigma}$ with $\C T$ additive and $\modulesfp{\C T}$ abelian. We will use this fact now. More precisely, in order to apply a result of André we will use that, under these hypotheses, any object in $\modulesfp{\C T_{\Sigma}}$ has a projective resolution by objects in $\C T_{\Sigma}$.
Since $y$ is tautologically the restriction of the identity functor in $\modulesfp{\C T_{\Sigma}}$ to $\C T_\Sigma$, and the identity functor is exact, then $H_*(M,y)$ is naturally $M$ concentrated in degree $0$, see \cite[Proposition 13.2]{andre_methode_1967}. Therefore the complex $\mathbb Z (M)\otimes_{\mathbb Z\C T_{\Sigma}}B_{\bullet}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma}$ is a projective resolution of $M$. In particular, since $\mathbb Z\modulesfp{\C T_{\Sigma}}$ is locally free by definition, the $E_2$-term of the first-vertical-then-horizontal spectral sequence is as in the statement.
André's result would in principle require that $\C T_\Sigma$ identified with the full subcategory of projectives in $\modulesfp{\C T_\Sigma}$ under the Yoneda inclusion. That would be true if $\C T$ were idempotent complete, which is a harmless common assumption in this paper. Nevertheless, it suffices that we can form a simplicial resolution of any finitely presented right $\C T_\Sigma$-module $M$ with objects in $\C T_\Sigma$, and by the Dold--Kan equivalence this follows from the existence of a projective resolution of $M$ by objects in $\C T_\Sigma$.
In order to compute the target of the previous spectral sequence of the bicomplex $C^{\star,\bullet}$ we consider the other one, exactly as in the proof of Proposition \ref{spectral_sequence}.
The $\mathbb Z\modulesfp{\C T_{\Sigma}}$-bimodule $\hom_{\C T_{\Sigma}}^*(\mathbb Z(-)\otimes_{\mathbb Z\C T_{\Sigma}}B_{n}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma},-))$ sends $M$ and $N$ to
\[\prod_{X_1,\dots,X_n}\hom^*_{\operatorname{Set}}(M(X_0)\times\C T_{\Sigma}(X_1,X_{0})\times\cdots\times\C T_{\Sigma}(X_n,X_{n-1}),N(X_n)),\]
hence, it is a product of $\mathbb Z\modulesfp{\C T_{\Sigma}}$-bimodules $D_{X,Y}$ of the form $D_{X,Y}(M,N)=\hom_{\operatorname{Set}}^*(M(X),N(Y))$, where $X$ and $Y$ are fixed objects in $\C T_{\Sigma}$. The cohomology of $\modulesfp{\C T_{\Sigma}}$ with coefficients in such a $D_{X,Y}$ is concentrated in degree $0$ by the graded version of \cite[Lemma 3.9]{jibladze_cohomology_1991} (which actually follows from the ungraded original version via the graded-ungraded long exact sequence). Hence, the inclusion of the $0$-dimensional horizontal cohomology in $C^{\star,\bullet}$ is a quasi-isomorphism (with the total complex). This $0$-dimensional horizontal cohomology is the end of the cochain complex of $\mathbb Z\modulesfp{\C T_{\Sigma}}$-bimodules $\hom_{\C T_{\Sigma}}^*(\mathbb Z(-)\otimes_{\mathbb Z\C T_{\Sigma}}B_{\bullet}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma},-))$.
By the extension-restriction of scalars adjunction, this complex coincides with $\hom_{\mathbb Z\C T_{\Sigma}}^*(\mathbb Z(-)\otimes_{\mathbb Z\C T_{\Sigma}}B_{\bullet}(\mathbb Z\C T_{\Sigma}),-))$. The end is, dimensionwise, the graded abelian group of natural transformations from the source to the target regarded as graded functors $\modulesfp{\C T_{\Sigma}}\rightarrow \modules{\C T_{\Sigma}}$. The source preserves colimits, and $\C T_\Sigma\subset\modulesfp{\C T_\Sigma}$ is the inclusion of a dense subcategory \cite[\S5.1]{kelly_basic_2005}, hence the source is the left Kan extension of its restriction along $\C T_\Sigma\subset\modulesfp{\C T_\Sigma}$ \cite[Theorem 5.29]{kelly_basic_2005}, so the end can be computed by restricting to $\C T_\Sigma$. The latter end is the complex $\hom_{\mathbb Z\C T_{\Sigma}^{\env}}^*(B_\bullet(\mathbb Z\C T_{\Sigma}),\C T_{\Sigma})$, whose cohomology is the claimed target of the spectral sequence, hence we are done. An explicit quasi-isomorphism $\xi\colon \hom_{\mathbb Z\C T_{\Sigma}^{\env}}^*(B_\bullet(\mathbb Z\C T_{\Sigma}),\C T_{\Sigma})\hookrightarrow C^{\star,\bullet}$ is defined as in \eqref{quasi_iso}.
\end{proof}
Now we can state the analogue of Theorem \ref{theorem_edge_morphism}.
\begin{theorem}\label{theorem_edge_morphism_top}
Let $\C T$ be an idempotent complete triangulated category with suspension $\Sigma$ and a topological enhancement. The edge morphism
\[\hbw{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\To \hbw{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}
\]
of the spectral sequence in Proposition \ref{spectral_sequence_top} takes the universal Toda bracket of the enhancement to the Toda bracket of the triangulated structure.
\end{theorem}
The proof is exactly the same as that of Theorem \ref{theorem_edge_morphism}, replacing the reference to Proposition \ref{spectral_sequence} with Proposition \ref{spectral_sequence_top}.
Despite this section's non-additive cohomology is based on cochains which are not multilinear, in certain cases we can compute it using cochains which at least vanish when evaluated at zero maps. More precisely, let $\C C$ be a graded category with a zero object $0$. A $\mathbb Z\C C$-bimodule $M$ is \emph{zero-trivial} if it vanishes when evaluated at the zero object of $\C C$ at any slot, $M(0,-)=0=M(-,0)$. If $\C C$ is additive, any $\C C$-bimodule regarded as a $\mathbb Z\C C$-bimodule is zero-trivial. A cochain $\varphi\in \hc{\star, \ast}{\mathbb Z\C C,M}$ is \emph{zero-normalized} if it vanishes whenever we put a trivial morphism in one of the slots $\varphi(\dots,0,\dots)=0$. The inclusion of the subcomplex $\hz{\star, \ast}{\mathbb Z\C C,M}\subset \hc{\star, \ast}{\mathbb Z\C C,M}$ consisting of zero-normalized cochains is a quasi-isomorphism when $M$ is zero-trivial, actually a chain homotopy equivalence, compare \cite[Theorem 1.10 and Appendix B]{baues_cohomology_1989}. The subcomplex of normalized cochains is better explained as follows. When $\C C$ has a zero object we can construct its \emph{zero-normalized linearization} $\breve{\mathbb Z}\C C$, with the same objects as $\C C$ and where $(\breve{\mathbb Z}\C C)(X,Y)$ is obtained from $\mathbb Z\C C(X,Y)$ by quotienting out the zero maps $0\in\C C(X,Y)$. There is an obvious projection linear functor $\mathbb Z\C C\rightarrow \breve{\mathbb Z}\C C$ which is the identity on objects. A zero-trivial $\mathbb Z\C C$-bimodule is the same as a $\breve{\mathbb Z}\C C$-bimodule $M$, and $\hz{\star, \ast}{\mathbb Z\C C,M}=\hc{\star, \ast}{\breve{\mathbb Z}\C C,M}$. The inclusion of zero-normalized cochains is induced by the projection $\mathbb Z\C C\rightarrow \breve{\mathbb Z}\C C$.
Given a graded linear category $\C C$ such that $\modulesfp{\C C}$ is Frobenius abelian. The same formula as in Definition \ref{kappa}, using now zero-normalized cochains, yields graded abelian group morphisms, $p\geq 1$, $q\in\mathbb Z$,
\[\kappa \colon \hbw{p+1,*}{\modulesfp{\C C },\widehat{\ext}_{\C C}^{q,*}}
\To
\hbw{0,*}{\modulesfp{\C C },\widehat{\ext}_{\C C}^{p+q,*}}.\]
The use of zero-normalized cochains ensures that the definition does not depend on the choice of representing cocycles.
The analogue of Proposition \ref{Heller_criterion_by_differential} holds in our current non-additive setting.
\begin{proposition}\label{Heller_criterion_by_differential_top}
If $\C T$ is a small ungraded additive category such that $\modulesfp{\C T}$ is Frobenius abelian and $\Sigma\colon \C T\rightarrow\C T$ is an automorphism, then the set of Puppe triangulated structures on $\C T$ with suspension functor $\Sigma$ is in bijection with the units of the bigraded ring $\hbw{0,*}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$ lying in the kernel of the composite
\begin{center}
\begin{tikzcd}
\hbw{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}
\arrow[d,"d_2"]\\
\hbw{2,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{2,*}}
\arrow[d,"\kappa"]\\
\hbw{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}
\end{tikzcd}
\end{center}
where $d_2$ is a second differential in the spectral sequence of Proposition \ref{spectral_sequence_top}.
\end{proposition}
The proof of this result is essentially the same as the proof of Proposition \ref{Heller_criterion_by_differential}. We only need to replace the resolution $M\otimes_{\C T_\Sigma}B_\bullet(\C T_\Sigma)$ of $M$ used previously with $\mathbb Z (M)\otimes_{\mathbb Z\C T_{\Sigma}}B_{\bullet}(\mathbb Z\C T_{\Sigma})\otimes_{\mathbb Z\C T_{\Sigma}}\C T_{\Sigma}$, that we use in the proof of Proposition \ref{spectral_sequence_top}. We did not really use in an essential way the multilinearity of cocycles therein. We only used that cocycles vanish when one variable is the trivial morphism. Hence, we must restrict to zero-normalized cochains, i.e.~we must actually use $\breve{\mathbb Z}\C C$ instead of $\mathbb Z\C C$ and we must define a zero-normalized forceful linearization $\breve{\mathbb Z}(M)$ of right $\C T_\Sigma$-modules $M$ where $\breve{\mathbb Z}(M)(X)$
is obtained from $\mathbb Z(M(X))$ by quotienting out zeroes $0\in M(X)$.
Theorem \ref{octahedral_theorem} (the sufficient condition for the octahedral axiom) also holds true for the spectral sequence in Proposition \ref{spectral_sequence_top}, so triangulated categories with a universal Toda bracket satisfy the octahedral axiom (this was already checked in \cite{baues_cohomologically_2008}).
\begin{theorem}\label{octahedral_theorem_top}
If $\C T$ is a small ungraded idempotent complete additive category such that $\modulesfp{\C T}$ is Frobenius abelian, $\Sigma\colon \C T\rightarrow\C T$ is an automorphism, and $\varphi\in \hbw{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}$ corresponds to a Puppe triangulated structure on $\C T$ with suspension $\Sigma$ such that $d_2(\varphi)=0$, then this Puppe triangulated structure satisfies the octahedral axiom.
\end{theorem}
Again, the proof of this theorem is not much different to its linear version over a field, up to the previous replacement of resolutions.
The topological analogue of DG- or $A$-infinity categories are \emph{spectral categories}, i.e.~categories enriched in any closed symmetric monoidal category of spectra, such as symmetric spectra. Given a spectral category $\C C$, we can take stable homotopy groups on morphism spectra and form a graded linear category $\pi_*\C C$. In the same way as a DG-category induces Massey products in cohomology, a spectral category $\C C$ induces Toda brackets in stable homotopy, hence if $\pi_*\C C \simeq \C T_\Sigma$ is weakly stable then $(\C T,\Sigma)$ is endowed with a stable Toda bracket. We say that $\C C$ is \emph{pre-triangulated} if in the previous circumstances the stable Toda bracket induces a triangulated structure on $(\C T,\Sigma)$. This stable Toda bracket comes from a universal Toda bracket. Indeed, we can consider the stable model category $\modules{\C C}$ of right $\C C$-modules, and the Yoneda inclusion $\C C\subset\modules{\C C}$ gives rise to a full inclusion $\pi_0\C C\subset\operatorname{Ho}\modules{\C C}$. If $\Sigma$ is the suspension functor in $\operatorname{Ho}\modules{\C C}$ and $\pi_*\C C$ is weakly stable then $\pi_*\C C=(\pi_0\C C)_{\Sigma}$, so it inherits the universal Toda bracket. Up to idempotent completion, our pre-triangulated spectral categories coincide with Tabuada's triangulated spectral categories \cite[Definition 5.1]{tabuada_matrix_2010}. The topological analogue of Corollary \ref{pretriangulated} is the following result.
\begin{corollary}\label{pretriangulated_top}
Let $\C C$ be a spectral category such that idempotents in $\pi_0\C C$ split. The following statements are equivalent:
\begin{itemize}
\item $\C C$ is pre-triangulated.
\item $\pi_*\C C$ is weakly stable, $\modulesfp{\pi_0\C C}$ is Frobenius abelian, and the image of the universal Massey product along the edge morphism \[\hbw{3,-1}{\pi_*\C C, \pi_*\C C}\longrightarrow \hbw{0,-1}{\modulesfp{\pi_*\C C},\ext_{\pi_*\C C}^{3,\ast}}\] in Theorem \ref{theorem_edge_morphism_top} is a unit in the bigraded algebra \[\hbw{0,*}{\modulesfp{\pi_*\C C},\widehat{\ext}_{\pi_*\C C}^{\star,\ast}}.\]
\end{itemize}
\end{corollary}
\section{Example of non-vanishing obstructions}\label{example}
We here illustrate with an example that the obstructions need not vanish. For this, we need a triangulated category without enhancements. There are few known examples, essentially those in \cite{muro_triangulated_2007}, their non-commutative analogues \cite{dimitrova_triangulated_2009}, and a new recent family of examples over the rationals \cite{rizzardo_k-linear_2018}. These new examples do not have an $A_\infty$-enhancement but they do have an $A_6$-enhancement, so they have a universal Massey product, and therefore the first obstructions described in this paper vanish. There are also some triangulated categories defined over a field, the non-standard finite $1$-Calabi-Yau triangulated categories \cite{amiot_structure_2007}, for which no enhancements are known, but their triangulated structures are also defined from universal Massey products.
We concentrate in the simplest example considered in \cite{muro_triangulated_2007}, the category $\C T=\free{\mathbb Z/4}$ of finitely generated free $\mathbb Z/4$-modules with the identity suspension functor $\Sigma=\operatorname{id}_{\C T}$. This category is idempotent complete since all projective $\mathbb Z/4$-modules are free. We showed that this category has a triangulated structure where
\[\mathbb Z/4\stackrel{2}\longrightarrow \mathbb Z/4\stackrel{2}\longrightarrow\mathbb Z/4\stackrel{2}\longrightarrow\mathbb Z/4\]
is an exact triangle. We will regard this category as a $\mathbb Z$-linear category and show that there must be a non-vanishing topological obstruction. Let us first place this triangulated structure within the abelian group of stable Toda brackets, see Theorem \ref{Toda_brackets_and_cohomology}.
\begin{proposition}
If $\C T=\free{\mathbb Z/4}$ and $\Sigma\colon\C T\rightarrow \C T$ is the identity functor, then the previous triangulated structure is the only existing triangulated structure on the pair $(\C T,\Sigma)$ and corresponds under the bijection in Corollary \ref{Heller_graded} to the non-trivial element in
\[\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}\cong\mathbb Z/2.\]
\end{proposition}
\begin{proof}
Right modules over $\free{\mathbb Z/4}$ are the same as $\mathbb Z/4$-modules. The stable module category $\modulesst{\C T}=\modulesst{\mathbb Z/4}$ is the category $\modulesfp{\mathbb Z/2}$ of finite-dimensional $\mathbb Z/2$-vector spaces since any finitely generated $\mathbb Z/4$-module is a finite direct sum of copies of $\mathbb Z/4$ and $\mathbb Z/2$. Using the short exact sequence
\[\mathbb Z/2\hookrightarrow\mathbb Z/4\twoheadrightarrow\mathbb Z/2 \]
it is easy to see that the cosyzygy functor $S$ is the identity, as $\Sigma$.
By Proposition \ref{Heller_ungraded},
\[\begin{split}
\hh{0}{\modulesfp{\C T},\ext_{\C T}^3(-,\Sigma^{-1})}&=\hh{0}{\modulesfp{\C T},\widehat{\ext}_{\C T_{\Sigma}}^{3,-1}}\\&\cong \hh{0}{\modulesst{\C T},\homst_{\C T}(\Sigma,S^{3})}
\end{split}\]
is the group of natural transformations $\Sigma\rightarrow S^3$ in $\modulesst{\C T}$, i.e.~the endomorphisms of the identity functor in $\modulesfp{\mathbb Z/2}$, which is $\mathbb Z/2$.
Since $\Sigma$ is the identity, the bottom map in the exact sequence \eqref{exact_cohomology_sequence} is multiplication by $2$. We have just seen that the source (and target) of this map is isomorphic to $\mathbb Z/2$. Hence $i^*$ is an isomorphism
\[\hh{0,-1}{\modulesfp{\C T_\Sigma},\ext_{\C T_{\Sigma}}^{3,*}}\cong \hh{0}{\modulesfp{\C T},\ext_{\C T}^3(-,\Sigma^{-1})}\]
and all Toda brackets are stable, see Theorem \ref{Toda_brackets_and_cohomology}.
Any triangulated structure on $(\C T,\Sigma)$ must correspond to the non-trivial element since it must be a unit in $\hh{0,*}{\modulesfp{\C T_\Sigma},\widehat{\ext}_{\C T_{\Sigma}}^{\bullet,*}}$, which is non-trivial. In particular, the previous triangulated structure on this pair is unique.
\end{proof}
Our strategy to prove that one of the obstructions must be non-vanishing will be to show that the no-trivial element is not in the image of the edge morphism in Theorem \ref{theorem_edge_morphism_top},
\[\hbw{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\To \hbw{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}\cong\mathbb Z/2.
\]
This indeed guarantees that some obstriction is not zero, but it does not say which one. We could compute the obstructions explicitly but that would take much longer (computing spectral sequence differentials is difficult) and there would not be a clear benefit. The computation goes through several steps, where we will use the following ungraded version of the spectral sequence in Proposition \ref{spectral_sequence_top}.
\begin{proposition}\label{spectral_sequence_ungraded}
If $\C T$ is a small additive category such that $\modulesfp{\C T}$ is abelian, then there is a first quadrant cohomological spectral sequence
\[E_2^{p,q}=H^{p}(\modulesfp{\C T },\ext_{\C T }^{p})\Longrightarrow H^{p+q}(\C T ,\C T ).\]
\end{proposition}
\begin{proof}
It is the spectral sequence of the bicomplex
\[\hom_{\mathbb Z\modulesfp{\C T }^{\env}}^*(B_{\star}(\mathbb Z\modulesfp{\C T }),\hom_{\C T }^*(\mathbb Z(-)\otimes_{\mathbb Z\C T }B_{\bullet}(\mathbb Z\C T )\otimes_{\mathbb Z\C T }\C T ,-)).\]
The proof is exactly the same as for Proposition \ref{spectral_sequence_top} since the hypotheses imply that any finitely presented right $\C T$-module has a projective resolution by objects in $\C T$.
\end{proof}
This proposition has a Hochschild analogue, but we will not use it here.
\begin{remark}\label{morphism_ss}
It is straightforward to notice, looking at the bicomplexes defining these spectral sequences, that the following graded-to-ungraded comparison morphisms
\begin{gather*}
i^*\colon H^{p,0}(\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{p,*})\longrightarrow H^{p}(\modulesfp{\C T },\ext_{\C T }^{p}),\\
i^*\colon H^{p+q,0}(\C T_{\Sigma},\C T_{\Sigma})\longrightarrow H^{p+q}(\C T ,\C T ),
\end{gather*}
which fit into the long exact sequence \eqref{exact_cohomology_sequence_top} derived from Proposition \ref{exact_cohomology_sequence},
are part of a morphism from the spectral sequence in Proposition \ref{spectral_sequence_top} to that in Proposition \ref{spectral_sequence_ungraded}.
\end{remark}
We will actually use a version with coefficients of the previous spectral sequence.
\begin{proposition}\label{spectral_sequence_with_coefficients}
If $\C T$ is a small additive category such that $\modulesfp{\C T}$ is abelian and $M$ is a $\C T$-bimodule, there is a first quadrant cohomological spectral sequence of graded abelian groups
\[E_2^{p,q}=H^{p}(\modulesfp{\C T },\ext_{\C T }^{p}(-,-\otimes_{\C T}M))\Longrightarrow H^{p+q}(\C T ,M ).\]
\end{proposition}
\begin{proof}
It is the spectral sequence of the bicomplex
\[\hom_{\mathbb Z\modulesfp{\C T }^{\env}}^*(B_{\star}(\mathbb Z\modulesfp{\C T }),\hom_{\C T }^*(\mathbb Z(-)\otimes_{\mathbb Z\C T }B_{\bullet}(\mathbb Z\C T )\otimes_{\mathbb Z\C T }\C T ,-\otimes_{\C{T}}M)).\]
Observe that this is the same bicomplex as in the proof of Proposition \ref{spectral_sequence_ungraded} with the last slot tensored by $M$. The proof is not any different to the proofs of Propositions \ref{spectral_sequence_top} and \ref{spectral_sequence_ungraded} since the last slot does not play any relevant role therein, it only slightly changes the $E_2$ term and the target.
\end{proof}
This proposition has graded and Hochschild analogues that will not be used in this paper. The particular version we present here is a special case of \cite[Theorem B]{jibladze_cohomology_1991}. The target in Jibladze and Pirashvili's spectral sequence is apparently different to ours, but it is possible to check that both coincide under our assumptions.
\begin{remark}\label{morphism_ss_coefficients}
As in Remark \ref{morphism_ss}, the following graded-to-ungraded comparison morphisms
\begin{gather*}
i^*\colon H^{p,r}(\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{p,*})\longrightarrow H^{p}(\modulesfp{\C T },\ext_{\C T }^{p}(-,\Sigma^{r})),\\
i^*\colon H^{p+q,r}(\C T_{\Sigma},\C T_{\Sigma})\longrightarrow H^{p+q}(\C T ,\C T(-,\Sigma^{r})),
\end{gather*}
also fitting into the long exact sequence \eqref{exact_cohomology_sequence_top} derived from Proposition \ref{exact_cohomology_sequence},
are part of a morphism from the spectral sequence in Proposition \ref{spectral_sequence_top} to that in Proposition \ref{spectral_sequence_with_coefficients} for $M=\C T_{\Sigma}^r=\C T(-,\Sigma^r)$.
\end{remark}
We can define the \emph{Hochschild homology} $HH_\star(\C C, M)$ of a graded $k$-linear category $\C C$ with coefficients in a $\C C$-bimodule $M$ as the homology of $B_\ast(\C C)\otimes_{\C C^{\env}}M$. Here $\star$ is the Hochschild grading but there is an extra hidden grading coming from the fact that $\C C$ is graded, as with cohomology. Moreover, if $\C C$ is non-linear, i.e.~enriched in graded sets, we can define the \emph{homology} $H_\star(\C C,M)$ of $\C C$ with coefficients in a $\mathbb Z\C C$-bimodule $M$ as $HH_\star(\mathbb Z\C C, M)$. The ungraded cases are just particular cases of this.
Let $R$ be a ring and $\free{R}$ the category of finitely generated free $R$-modules. The homology $H_\star(\free{R},\hom_R)$ is the Mac Lane or topological Hochschild homology of $R$ \cite{jibladze_cohomology_1991,pirashvili_mac_1992}. We need yet another spectral sequence, this time a universal coefficients one.
\begin{proposition}\label{universal_coefficients}
For any commutative ring $R$ and any $R$-module $M$, there is a first quadrant cohomological spectral sequence
\[E_2^{p,q}=\ext^p_R(H_q(\free{R},\hom_R),M)\Longrightarrow H^{p+q}(\free{R},\hom_R(-,-\otimes_RM)).\]
\end{proposition}
\begin{proof}
Given two finitely generated free $R$-modules $P$ and $Q$, there are natural isomorphisms
$$\hom_R(P,Q\otimes_R M)\st{\alpha}\longleftarrow \hom_R(P,Q)\otimes_RM\st{\beta}\To\hom_R(\hom_R(Q,P),M)$$
defined by
\begin{align*}
\alpha(f\otimes m)(x)&=f(x)\otimes m\\
\beta (f\otimes m)(g)&=\operatorname{trace}(fg)\cdot m.
\end{align*}
Indeed the morphisms $\alpha$ and $\beta$ are clearly well defined and natural, so it is enough to check that they are isomorphisms for $P=Q=R$, and this case is trivial.
The isomorphisms $\alpha$ and $\beta$ define an isomorphism of cochain complexes
\begin{multline*}
\hom_{\mathbb Z\free{R}^{\env}}(B_\star(\mathbb Z\free{R}),\hom_R(-,-\otimes_RM))\\
\cong \hom_R(B_\star(\mathbb Z\free{R})\otimes_{\mathbb Z\free{R}} \hom_R,M).
\end{multline*}
Indeed, dimension-wise these complexes look like
\begin{gather*}
\prod_{P_0,\dots,P_n}\hom_{\mathbb Z}\left(\bigotimes_{i=1}^n \mathbb Z\hom_R(P_i,P_{i-1}), \hom_R(P_n,P_0\otimes M)\right),\\
\prod_{P_0,\dots,P_n}\hom_{R}\left(\bigotimes_{i=1}^n \mathbb Z\hom_R(P_i,P_{i-1})\otimes \hom_R(P_0,P_n), M\right),
\end{gather*}
where the $P_i$ run over all sequences of $n$ finitely generated free $R$-modules. Hence, the top module is a product of copies of $\hom_R(P_n,P_0\otimes M)$ and the bottom module, by adjunction, is a product of copies of $\hom_R(\hom_R(P_n,P_0),M)$. Both products are indexed by the same set, which is the union of all sets $\prod_{i=1}^n \hom_R(P_i,P_{i-1})$ over all possible choices of $P_0,\dots, P_n$, so we just apply the isomorphism $\beta\alpha^{-1}$ factorwise. Compatibility with differentials is straightforward.
If $I^*$ is an injective resolution of $M$, then the spectral sequence of the statement is the first-horizontal-then-vertical cohomology spectral sequence of the bicomplex
\[\hom_R(B_\star(\mathbb Z\free{R})\otimes_{\mathbb Z\free{R}} \hom_R,I^*).\]
The identification of the $E_2$-term is obvious because the functors $\hom_R(-,I^n)$ are exact. For the target of this spectral sequence, we use the other one, obtained by first taking vertical homology and then horizontal homology. As we have seen above, $B_\star(\mathbb Z\free{R})\otimes_{\mathbb Z\free{R}} \hom_R$ consists of free $R$-modules, so the vertical homology of the bicomplex is $\hom_R(B_\star(\mathbb Z\free{R})\otimes_{\mathbb Z\free{R}} \hom_R,M)$ concentrated in vertical degree $0$. Hence we are done.
\end{proof}
\begin{corollary}
For any $\mathbb Z/4$-module $M$ we have a natural isomorphism \[H^{3}(\free{\mathbb Z/4},\hom_{\mathbb Z/4}(-,-\otimes_{\mathbb Z/4}M))\cong \hom_{\mathbb Z/4}(\mathbb Z/2,M).\]
\end{corollary}
\begin{proof}
The topological Hochschild homology of $\mathbb Z/4$ is computed in \cite{brun_topological_2000}. The lower homology groups are
\begin{equation*}
H_n(\free{\mathbb Z/4},\hom_{\mathbb Z/4}) \;\;\cong\;\;\left\{\begin{array}{ll}
\mathbb Z/4,&n=0,\\
0,&n=1,\\
\mathbb Z/4,&n=2,\\
\mathbb Z/2,&n=3.
\end{array}\right.
\end{equation*}
This implies that the $E_2$-term of the universal coefficients spectral sequence in Proposition \ref{universal_coefficients} satisfies $E^{p,q}_2=0$ for $p>0$ and $q<3$. Therefore
\[
H^{3}(\free{\mathbb Z/4},\hom_{\mathbb Z/4}(-,-\otimes_{\mathbb Z/4}M))\cong E_2^{0,3}=
\hom_{\mathbb Z/4}(\mathbb Z/2,M).
\]
\end{proof}
\begin{corollary}\label{trivial}
The natural projection $\mathbb Z/4\twoheadrightarrow \mathbb Z/2$ induces the trivial morphism
\[H^{3}(\free{\mathbb Z/4},\hom_{\mathbb Z/4}) \stackrel{0}\longrightarrow
H^{3}(\free{\mathbb Z/4},\hom_{\mathbb Z/4}(-,-\otimes_{\mathbb Z/4}\mathbb Z/2)).\]
\end{corollary}
\begin{proposition}\label{injective}
The natural projection $\mathbb Z/4\twoheadrightarrow \mathbb Z/2$ induces an injective morphism
\[H^0(\modulesfp{\mathbb Z/4},\ext^n_{\mathbb Z/4})\hookrightarrow
H^0(\modulesfp{\mathbb Z/4},\ext^n_{\mathbb Z/4}(-,-\otimes_{\mathbb Z/4}\mathbb Z/2)).\]
\end{proposition}
\begin{proof}
An element $\varphi$ in the source is a family of elements
\[\varphi(N)\in \ext^n_{\mathbb Z/4}(N,N),\]
where $N$ runs over all finitely generated $\mathbb Z/4$-modules, and similarly for the target. Any $N$ decomposes as a direct sum of copies of $\mathbb Z/4$ and $\mathbb Z/2$. Hence, by \cite{baues_sum-normalised_1996}, $\varphi$ only depends on the two special values $\varphi(\mathbb Z/4)$ and $\varphi(\mathbb Z/2)$, and the former is zero because $\mathbb Z/4$ is projective, so the latter suffices. Now the result follows from the fact that the natural projection obviously induces an injective morphism (actually an isomorphism)
\[\ext^n_{\mathbb Z/4}(\mathbb Z/2,\mathbb Z/2\otimes_{\mathbb Z/4} \mathbb Z/4)\longrightarrow \ext^n_{\mathbb Z/4}(\mathbb Z/2,\mathbb Z/2\otimes_{\mathbb Z/4} \mathbb Z/2).\]
\end{proof}
We finally prove the result which implies that not all our first obstructions vanish, and hence $\free{\mathbb Z/4}$ does not have any topological enhancement.
\begin{proposition}
If $\C T=\free{\mathbb Z/4}$ and $\Sigma\colon\C T\rightarrow \C T$ is the identity functor, then the edge morphism
\[\hbw{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\To \hbw{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}
\]
of the spectral sequence in Proposition \ref{spectral_sequence_top} is trivial.
\end{proposition}
\begin{proof}
Recall that $\C T$ is idempotent complete, as required by Proposition \ref{spectral_sequence_top}, since projective $\mathbb Z/4$-modules are free, and $\C T$-modules are the same as $\mathbb Z/4$-modules.
Using the spectral sequence morphism in Remark \ref{morphism_ss_coefficients} we obtain a commutative diagram
\begin{center}
\begin{tikzcd}
\hbw{3,-1}{\C T_{\Sigma}, \C T_{\Sigma}}\arrow[r, "i^*"]\arrow[d]& \hbw{3}{\C T , \C T(-,\Sigma^{-1}) }\arrow[d]\\
\hbw{0,-1}{\modulesfp{\C T_{\Sigma}},\ext_{\C T_{\Sigma}}^{3,\ast}}\arrow[r, hook, "i^*"]& \hbw{0}{\modulesfp{\C T },\ext_{\C T }^{3}(-,\Sigma^{-1})}
\end{tikzcd}
\end{center}
where the vertical arrows are edge morphisms and the horizontal arrows are graded-to-ungraded comparison morphisms. The one at the bottom is injective by \eqref{exact_cohomology_sequence_top}. Therefore it suffices to show that the right edge morphism vanishes. The suspension functor is the identity, hence the right edge morphism is the morphism on the left of the following commutative diagram
\begin{center}
\begin{tikzcd}
\hbw{3}{\free{\mathbb Z/4}, \hom_{\mathbb Z/4}}\arrow[d]\arrow[r,"0"]&\hbw{3}{\free{\mathbb Z/4}, \hom_{\mathbb Z/4}(-,-\otimes_{\mathbb Z/4}\mathbb Z/2)} \arrow[d]\\
\hbw{0}{\modulesfp{\mathbb Z/4},\ext_{\mathbb Z/4}^{3}}
\arrow[r,hook]& \hbw{0}{\modulesfp{\mathbb Z/4},\ext_{\mathbb Z/4}^{3}(-,-\otimes_{\mathbb Z/4}\mathbb Z/2)}
\end{tikzcd}
\end{center}
This diagram is defined by the natural projection $\mathbb Z/4\twoheadrightarrow\mathbb Z/2$ and by the naturality in $M$ of the spectral sequence in Proposition \ref{spectral_sequence_with_coefficients}. The top arrow is trivial by Corollary \ref{trivial} and the bottom arrow is injective by Proposition \ref{injective}, therefore the left vertical arrow is necessarily trivial.
\end{proof}
We would like to remark that the computations carried out in this section are closely related to (some of them even directly taken from) the first (unpublished) proof of the fact that the triangulated structure of $\free{\mathbb Z/4}$ does not admit topological enhancements \cite{muro_triangulated_2007-1}.
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\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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| 166,390
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Industry veteran Paul Jones has been named as the new VP sales, Asia-Pacific for systems integrator Megahertz.
Industry veteran Paul Jones joins the company as VP sales, Asia-Pacific
Systems integrator (SI) Megahertz has appointed Paul Jones as VP sales, Asia-Pacific.
Jones joins Megahertz from Oracle’s Asian operations, where, as sales director, he was responsible for the very first Oracle cloud archive deal in the region and held the position as chair of the IABM regional council for APAC. Prior to that, he headed sales for the region at Front Porch Digital as the company’s first APAC employee, working with global brands such as Disney, Sony Pictures and Turner, and with leading regional broadcasters such as RTM, Media Prima, Astro, Mediacorp and Seven Network.
Leveraging Megahertz’s 30-plus years as a trusted global SI in the media and entertainment industries, Jones is confident that he can present Asia-Pacific broadcasters with the “very best” options when it comes to rolling out both established and new technologies. These include providing guidance and support to those transitioning from baseband to IP-based operations, 4K/Ultra HD (UHD) operations and file-based workflows, as well as traditional SDI infrastructures.
Jones said: “There has been a general decline in broadcast advertising revenue in the South-east Asian market, in particular, due to the surge in competition from global over-the-top (OTT)/video-on-demand (VoD) platforms.
“We’re keen to help them enhance their workflow to keep ahead of that curve, by implementing imaginative design and integration that work to smarter business models, generate engaging content cost-effectively and use more efficient multi-platform distribution systems.”
| 387,297
|
TITLE: How many $4$ digit numbers can be created from $0$, $3$, $4$, $6$, $7$, $9$ under the following conditions?
QUESTION [0 upvotes]: How many $4$ digit numbers can be created from digits: $0$, $3$, $4$, $6$, $7$, $9$?
I got this as:
$$\dfrac{6!}{2!} = 360$$
To exclude numbers starting with zero: $$\dfrac {5!}{2!} = 60$$
Final result therefore: $$360 - 60 = 300$$
Is this correct?
There are some additional tasks I can't figure out:
a) In how many of them no digit is present more than once?
b) How many of them contain only one digit three times?
c) How many of them are even?
d) How many of them can be divided by $3$ with digit sum of $18$?
REPLY [0 votes]: Units place= 6 different possibilities
Tens-place= 6 different possibilities
hundreds place= 6 different possibilities
thousands place =5 different possibilities (can't be zero)
So number of different number is the product
6x6x6x5= 1080
| 217,428
|
TITLE: The limit $\lim_{n\to\infty}\dfrac{\ln(n+1) \sin(n)}{n \tan(n)}$
QUESTION [0 upvotes]: $$\lim_{n\to\infty}\dfrac{\ln(n+1) \sin(n)}{n \tan(n)}$$
No idea what to do with this one! Have tried writing out as series but that method is extremely complicated. L'hopitals doesn't work here either
REPLY [1 votes]: Hint: Use squeeze theorem $$0\le \left| \frac { \ln { \left( n+1 \right) \sin { \left( n \right) } } }{ n\tan { \left( n \right) } } \right| =\left| \frac { \ln { \left( n+1 \right) \cos { \left( n \right) } } }{ n } \right| =\left| \frac { \ln { \left( n+1 \right) } }{ n } \right| \left| \cos { \left( n \right) } \right| \le \left| \frac { \ln { \left( n+1 \right) } }{ n } \right| $$
| 1,507
|
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| 227,563
|
TITLE: Short Five Lemma for Fibrations
QUESTION [1 upvotes]: Is there a short five lemma for fibrations in algebraic topology (in whatever category where it would be suitable -- the topological category, the homotopy category, whatever).
By short five lemma I mean as follows. Let $E \to B$ is a fibration, with fiber $F$, and $E' \to B$ be another fibration with fiber $F'$. Suppose there are maps from $F \to F'$, $E \to E'$, and $B \to B'$ that all satisfy the obvious commutative diagram. Suppose all of the spaces are connected. If the maps from $F$ and $B$ are isomorphisms in the appropriate category, is the map from $E$ an isomorphism?
REPLY [1 votes]: What do you mean by the map between the fibres $F \to F'$ ? The projections $E \to B$ and $E' \to B'$ have lots of fibres. Do you mean that the map $\phi \colon E \to E'$ sends every fibre of $E \to B$ to a fibre of $E' \to B'$ and on each one is an isomorphism or do you just mean it sends one fibre in that way ?
If the former and you are interested in manifolds and isomorphisms are diffeomorphisms the result is true. A proof would be to notice that the map $E \to E'$ is a bijection because it is a bijection on fibres and the map on the base is a bijection. Hence there is a set-theoretic inverse. If $e \in E$ then the condition of being a diffeomorphism on the fibre through $e$ and on the base will be enough to show that the derivative of this map is a bijection. Hence the inverse function theorem shows there is a smooth local inverse which must coincide with the global set-theoretic inverse.
Not sure about the topological category with homeomorphisms as isomorphisms. Does that follow from mland's answer ? Is a weak homotopy inverse which is a bijection a homeomorphism ?
| 179,653
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‘Dreamgirls’ takes center stage at Stranahan TheaterWritten by Matt Liasse | | mliasse@toledofreepress.com
Charity Dawson is starring in her favorite musical. The Tony and Academy Award-winning “Dreamgirls” will play at the Stranahan Theater from April 25-28. Dawson plays Effie in the role of her lifetime.
“I do not just say that,” Dawson said. “It is my favorite musical of all time. I didn’t care if I was just in the back; I just wanted to be in the show.”
The show follows the backstage drama of a 1960s girl groupand was inspired by the career of Diana Ross and The Supremes, according to a news release. The Broadway hit includes songs “And I Am Telling You I’m Not Going” and “Listen,” with book and lyrics by Tom Eyen.
The local show is directed by Robert Longbottom and produced by Big League Productions.
Dawson heard the original cast recording of “Dreamgirls” as a child. She saw it on stage in 2006 at the Prince Music Theater in Philadelphia.
“I love it; I saw it nine times there,” she said.
She said she loves how “Dreamgirls” is written and the way it takes the audience “backstage.” She said it is different from anything she has seen or heard before.
“I love the fact that it’s a very cinematic show,” Dawson said. “I fell in love with the writing and how the story was told and the music.”
She said she has been preparing for this role for a long time, since being assigned songs to sing from “Dreamgirls” when she was in school. As a result, she didn’t feel intimidated by the part.
“I have an understanding of the character,” she said. “I’ve been studying this show before I was even in it. It’s just a big role to take on and big shoes to fill. I just took it as it’s my turn.”
Dawson said although she enjoys the film adaptation featuring Jennifer Hudson playing Effie, she hasn’t watched the movie since getting the part. She wanted to approach the role on her own, without outside bias.
“I think every actor should find their truth in [a role] and get their own understanding of the character,” she said.
Dawson graduated from the American Musical and Dramatic Academy in New York in 2005. She said she knows the area well because her brother, Trinity Dawson, played football for the University of Toledo.
Showtimes are 8 p.m. April 25-27, 2 p.m. April 27-28 and 7 p.m. April 28. Tickets can be purchased starting at $23 at the Stranahan Theater Box Office or by phone at (419) 381-8851.
Tags: Diana Ross, dreamgirls, Stranahan Theater, The Supremes
| 323,545
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By Wee Li Shyen
Diploma in Creative Writing for Television & New Media
Singapore Polytechnic
A photo of Sun Yat Sen in his family home was the inspiration for Tjio Kayloe’s first book
(Photo of Tjio Kayloe by Wee Li Shyen)
An aged photo sparked one man’s inspiration and kicked off a five-year journey to tell the story of the founding father of the Republic of China.
Tjio Kayloe’s route to being an author was not a conventional one.
Now 71, Kayloe made his career in investment banking in Hong Kong and New York before settling down in Singapore in 1990 to start a publishing business for financial institutions. After his retirement 10 years ago, he got bored of hitting golf balls and decided to pick up a pen and pursue something that had always been at the back of his mind – writing a book.
Creative inspiration
The inspiration for the subject went back to his childhood. As a boy, he had always wondered about the portrait of a Chinese man hanging prominently beside his grandfather’s in his family home in Indonesia.
“When you’re a kid of four years old, of course you understand who your grandfather is. But what’s this other joker doing there?” Kayloe chuckled.
It was only much later that he realised that that man was Sun Yat-Sen, the first president of the Chinese Republic, and that his grandfather was a loyal supporter. This laid the seeds for Kayloe’s first book: The Unfinished Revolution: Sun Yat-Sen and the Struggle for Modern China.
Interestingly, Kayloe has a different perspective of Dr. Sun, who many consider a visionary hero.
“Here was a man who never succeeded in anything,” said Kayloe, “You could even say he was a total failure in life. His moment of glory lasted just two months, when he became provisional president, and he was in that seat for like a month and a half max. That was his moment of glory. The rest of his life was failure all the way until the day he died. And yet after his death, he became so revered, so well-known.”
Unlike most books on Dr. Sun, which have been written from the Chinese perspective, Kayloe’s take was on the time the revolutionary leader spent in Southeast Asia, which has not been written about much.
The book was shortlisted for the Creative Non-Fiction category of the 2018 Singapore Literature Prize.
Learning from the first attempt
Kayloe is currently working on his second book, which is about the warlords of Republican China. He is still uncertain how much beyond 1929 to include, as most accounts of the period end in 1928. But he’s confident he’ll complete this faster than his first work.
“I think my second book can be done in two years, maybe two and a half. Some might say I’ve learned the tricks of the trade,” he said.
When writing his first book, he encountered “several disasters” when files in his computer became corrupted and he had to redo the work. Fortunately he was able to recover the information through the hard copies he had kept as well as periodic backups he had made. Kayloe has now devised a system to save his files so that even if they are lost, he won’t have to start from scratch again.
Despite becoming savvier about the mechanics of the craft, Kayloe doesn’t believe he is the right person to give tips to young writers.
“I’m not in a position to give advice,” Kayloe wryly commented, “maybe by the time I write my third book.”
However, he shared that there has to be proper planning. He admits that he made the mistake of jumping into big ideas and then having to backtrack, thus taking longer to finish the work. Most importantly, “you must love your idea,” he declared, “because if you don’t enjoy it, it becomes drudgery.”
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\begin{document}
\begin{abstract}
In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions.
The symmetry considered is reflection through the origin.
Given an unimodular irreducible Pisot substitution, we consider the intersection of its
Rauzy fractal with the Rauzy fractal of the reverse substitution.
This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.
\end{abstract}
\maketitle
\section{Introduction}\label{sec:intro}
The Rauzy fractal is an important object in the study of dynamical systems associated
with the Pisot substitutions, in particular it plays a fundamental role in the study of the Pisot conjecture.
Geometrical and topological properties of Rauzy fractals have been studied extensively,
see among other references~\cite{arnoux-ito,canterini-siegel,holton-zamboni,pytheas_fogg,rauzy,siegel-thuswaldner,sirvent-wang}.
Symmetries in Rauzy fractals were studied in~\cite{sirvent:2},
in relation to symmetries that exhibit the symbolic languages
which define the Rauzy fractals.
In the present paper we continue the study the symmetric structure of these sets.
We consider the Rauzy fractal of a unimodular irreducible substitution and
the fractal of its reverse substitution,
in section~\ref{sec:substitutions} we give definitions of these objects.
We show that the intersection of these two sets is invariant under reflection through the origin (Corollary~\ref{cor:symmetricfractal}).
Later we show that this set, is obtained by running the balanced pair algorithm of the original substitution and its reverse substitution (Theorem~\ref{thm:main}).
The balanced pair algorithm was introduced by Livshits~(\cite{livshits}) in the context of the Pisot conjecture, it was also used in ~\cite{sirvent-solomyak} in the same context.
A variant of this algorithm was used later by the first author in~\cite{sellami:1,sellami:2},
in the study of the intersection of Rauzy fractals associated with
different substitutions having the same incidence matrix.
This version of the balanced pair algorithm is used in the present article,
we describe it in section~\ref{sec:bpa}.
The intersection of Rauzy fractals of substitutions having the same incidence matrix,
has been studied previously in~\cite{sing-sirvent}.
In section~\ref{sec:examples} we present some examples, in particular a well known family of Pisot substitutions (Example 2).
We describe the balanced pair algorithm in detail for these examples and the intersection of the corresponding Rauzy fractals.
We end the paper with a section of open problems and remarks.
\section{Substitutions and Rauzy fractals}\label{sec:substitutions}
A substitution on a finite alphabet $\A=\{1,\ldots,k\}$ is a map $\s$ from
$\A$ to the set of finite words in $\A$, i.e., ${\A}^{*}=\cup_{i\geq 0}{\A}^{i}$.
The map $\sigma$ is extended to $\A^*$ by concatenation, i.e.,
$\s(\emptyset)=\emptyset$ and $\s(UV)=\s(U)\s(V)$, for all $U$, $V\in\A^*$.
Let $U$ be a word in $\A$, we denote by $|U|$ the length of $U$.
We denote by
$[\sigma(i)]_j$ the $j$-th symbol of the word $\sigma(i)$, i.e.,
$\sigma(i)=[\sigma(i)]_1\cdots[\sigma(i)]_{|\sigma(i)|}$.
\smallskip
Let $A^{\N}$ (respectively $A^{\Z}$) denote the set of one-sided
(respectively two-sided) infinite sequences in $\A$.
The map $\sigma$, is extended to $\A^{\N}$ and $\A^{\Z}$ in the obvious way:
Let $u=\ldots u_{-1}\dot{u_0}u_1\ldots$ be an element of $\A^{\Z}$, where the dot is used to denote the zeroth position.
So $\s(u)$ is of the form:
$$
\cdots[\s(u_{-1})]_{1}
\cdots[\s(u_{-1})]_{|\s(u_{-1})|}
[\dot{\s(u_0)}]_1\cdots[\s(u_0)]_{|\s(u_0)|}[\s(u_1)]_1\cdots.
$$
\smallskip
We call $u\in\A^{\N}$ (or $u\in\A^{\Z}$) a {\em fixed point} of $\s$
if $\s(u)=u$ and {\em periodic} if there exists $l>0$ so that it is fixed for $\sigma^l$.
\smallskip
We write $l_i(U)$ for the number of occurrences of the symbol
$i$ in the word $U$ and denote the vector ${\bf l}(U)=(l_1(U),\ldots,l_k(U))^{t}$.
The {\em incidence matrix} of the substitution $\s$ is defined as the matrix
$M_{\s}=M=(m_{ij})$ whose entries $m_{ij}=l_i(\s(j))$ , for $1\leq i,j \leq k$.
Note that $M_{\s}({\bf l}(U))={\bf l}(\s(U))$, for all $U\in\A^*$.
We say the substitution is {\em primitive} if its incidence matrix is
primitive, i.e., all the entries of $M^r$ are positive for some $r>0$.
\smallskip
For a primitive substitution there are a finite number of periodic points.
So we shall assume the substitution has always a fixed point, since we can replace the substitution by a suitable power.
Let $u$ be fixed point of $\sigma$,
we consider the dynamical system $(\Omega_u,S)$,
where $S$ is the shift map on $\A^{\N}$ (respectively on $\A^{\Z}$) defined by
$S(v_0v_1\cdots)=v_1\cdots$ (respectively $S(v)=w$,
where $w_{i}=v_{i+1}$) and $\Omega_u$ is the closure, in the product topology, of the orbit
of the fixed point $u$ under the shift map $S$.
\smallskip
A {\em Pisot number} is a real algebraic integer greater than $1$ such that its Galois conjugates are of norm smaller than $1$.
The Pisot numbers are also known in the literature as {\em Pisot-Vijayaraghavan} or {\em PV numbers}.
We say that a substitution is Pisot if the Perron-Frobenius eigenvalue of the the incidence matrix is a Pisot number.
A substitution is {\em irreducible Pisot} if it is Pisot and the characteristic polynomial of the incidence matrix is irreducible.
An irreducible Pisot substitution is primitive~\cite{canterini-siegel}.
A substitution is {\em unimodular } if the absolute value of the determinant
of its incidence matrix is $1$.
\smallskip
There is a long standing conjecture that the dynamical system
associated with a unimodular irreducible Pisot substitution is measurably
conjugate to a translation on a $(k-1)$-dimensional torus
({\em cf.}~\cite{rauzy,solomyak,vershik-livshits}).
This conjecture is known in literature as the Pisot conjecture.
G. Rauzy approached it via geometrical realization of the symbolic system.
He proved it in the case of the tribonacci substitution,
$\s(1)=12$, $\s(2)=13$ and $\s(3)=1$ ({\em cf.}~\cite{rauzy}).
In his proof, the construction of a set in $\R^2$, in general $\R^{k-1}$, plays an important role.
This set is known as the Rauzy fractal associated with the substitution.
For references on conditions under which the Pisot conjecture is true,
among other references see~\cite{arnoux-ito,barge-diamond,barge-kwapisz,CANT,canterini-siegel,LMS,livshits,pytheas_fogg,rauzy,sing,sirvent-solomyak,sirvent-wang,solomyak,vershik-livshits}.
Before we define Rauzy fractals, we have to introduce some constructions and notation.
\smallskip
Let $\s$ be a unimodular Pisot substitution and
$\lambda$ the Perron-Frobenius eigenvalue of the incidence matrix $M$,
so $\lambda$ is a Pisot number.
The characteristic polynomial of $M$ might be reducible, so algebraic degree of
$\lambda$ is smaller or equal that $k$, the cardinality of the alphabet $\A$.
We decompose $\R^k$ into a direct sum of subspaces, determined by the eigenvualues of $M$.
In particular, we consider:
\begin{itemize}
\item Let $E^u$ be the {\em expanding space}, i.e. the $\lambda$-expanding space of $M$,
the eigen-space associated with the eigenvalue $\lambda$.
\item
Let $E^s$ be the {\em contracting space}. i.e. the $\lambda$--contracting space of $M$, the direct sum of
the eigen-spaces associated with the Galois conjugates of $\lambda$.
\item
Let $E^c$ be the {\em complementary space}, i.e. the direct sum of
the eigen-spaces associated with the remaining eigenvalues of $M$.
\end{itemize}
So, by the definition of the subspaces, we have
$\R^k=E^u\oplus E^s\oplus E^c$.
The space $E^c$ is trivial if and only if the substitution is irreducible.
Let $\pi:\R^k\ra E^s$ be the projection of $\R^k$ onto $E^s$ along
$E^u\oplus E^c$.
\smallskip
Let $u=\ldots u_{-2}u_{-1}u_0u_1u_2\ldots$ be a fixed point of $\sigma:\A^{\Z}\ra\A^{\Z}$,
consider the polygonal line or stepped line $(L_n)_{n\in\Z}$ on $\R^k$, given by
$$
L_n:=\left\{\begin{array}{lc}
\sum_{i=0}^n e_{u_i} &\text{ if } n\geq 0\\
& \\
\sum_{i=n-1}^{-1} -e_{u_{i}} &\text{ if } n<0,
\end{array}\right.
$$
where $\{e_1,\ldots,e_k\}$ is the canonical basis of $\R^k$.
We define the {\em Rauzy fractal} associated with $\s$, as
$$
\RRR_{\sigma}:=\overline{\left\{\pi(L_n)\,|\, n\in\Z \right\}}.
$$
We shall also use the notation $\RRR$ for $\RRR_{\sigma}$, whenever the context is clear.
If we consider $\overline{\left\{\pi(L_n)\,|\, n\in\N \right\}}$,
we obtain the same set, ({\em cf.}~\cite{canterini-siegel});
see Figure~\ref{fig:projection}.
The construction of the Rauzy fractal,
does not depend on the selection of the fixed or periodic point of $\sigma$ ({\em cf.}~\cite{ pytheas_fogg}).
\smallskip
The Rauzy fractals are bounded~(\cite{holton-zamboni}),
they are the closure of their interior~(\cite{sirvent-wang}).
See~\cite{siegel-thuswaldner}, for a study of different topological properties of these sets.
The reducible case is studied in~\cite{EIR}.
\begin{figure}
\begin{center}
\scalebox{0.6}{\input{projection3.pstex_t} }
\caption{\label{fig:projection}Polygonal line projection in the Rauzy fractal construction for a substitution with $2$ symbols.
}
\end{center}
\end{figure}
\medskip
Let $\hat{\sigma}:\A\ra\A^*$ be the {\em reverse substitution of} $\sigma$, defined as follows:
$$[\hat{\sigma}(i)]_{j}=[\sigma(i)]_{|\sigma(i)|-j+1}.$$
If $\sigma$ is the tribonacci substitution: $1\ra12$, $2\ra13$, $3\ra 1$, then
$\hat{\sigma}$ is $1\ra 21$, $2\ra 31$, $3\ra 1$.
\smallskip
\begin{prop}\label{prop:FP}
Let $\sigma$ be a substitution such that it has a fixed point $u=\ldots u_{-1}u_0u_1\ldots$.
Let $\hat{\sigma}$ be the reverse substitution of $\s$.
Then $\hat{\sigma}$ has a fixed point,
$\hat{u}=\ldots\hat{u}_{-1}\hat{u}_0\hat{u}_1\ldots$,
with the property $\hat{u}_i=u_{-i-1}$, for $i\in\Z$.
\end{prop}
\begin{proof}
Let $u=\ldots u_{-1}\dot{u_0}u_1\ldots$ be a fixed point of $\s$, where the dot is used to denote the zeroth position.
So $u$ is of the from:
$$
\cdots[\s(u_{-1})]_{1}
\cdots[\s(u_{-1})]_{|\s(u_{-1})|}
[\dot{\s(u_0)}]_1\cdots[\s(u_0)]_{|\s(u_0)|}[\s(u_1)]_1\cdots.
$$
Let $v\in\A^{\Z}$ defined by $v_i=u_{-i-1}$, for $i\in\Z$, so $v$ is of the form
$$
\cdots[\s(u_0)]_{|\s(u_0)|}\cdots[\s(u_0)]_{1}[\dot{\s(u_{-1})}]_{|\s(u_{-1})|}[\s(u_{-1})|_{|\s(u_{-1})|-1}\cdots
[\s(u_{-1})]_1\cdots.
$$
By the definition of the reverse substitution $\hat{\s}$:
$$
v=\cdots\hat{\s}(u_0)\hat{\s}\dot{(u_{-1})}\hat{\s}(u_{-2})\cdots.
$$
Hence $v=\hat{\s}(v)$, i.e., $v$ is a fixed point for $\hat{\s}$.
\end{proof}
\smallskip
\begin{prop}
Let $\widehat{L}_n$ be the polygonal line associated with
the reverse substitution of $\sigma$. Then for all integer $n$ we have:
$$
\pi(L_n)=-\pi(\widehat{L}_{-n}).
$$
\end{prop}
\begin{proof}
By definition, $\widehat{L}_n=\sum_{i=0}^n e_{\hat{u}_i}$, if $n\geq 0$.
Due to Proposition~\ref{prop:FP}
$$\widehat{L}_n=\sum_{i=0}^n e_{u_{-i-1}}=\sum_{i=-n-1}^{-1} e_{u_{i}}=-L_{-n}.
$$
Similarly for $n<0$.
Since the projection $\pi$ is linear, we have
$\pi(L_{-n})=-\pi(\widehat{L}_n)$.
\end{proof}
\smallskip
The incidence matrices of $\sigma$ and $\hat{\sigma}$ are the same,
so both substitutions have the same spectral properties~({\em cf.}~\cite{queffelec}).
Therefore we can define the Rauzy fractal associated with $\hat{\sigma}$,
since $\sigma$ is a unimodular irreducible Pisot substitution.
From the previous Proposition follows the next result:
\begin{cor}\label{cor:symmetricfractal}
Let $\s$ be a unimodular irreducible Pisot substitution
and $\hat{\s}$ its reverse substitution.
Let $\RRR_{\s}$ and $\RRR_{\hat{\s}}$ be the corresponding Rauzy fractals.
Then $\RRR_{\s}=-\RRR_{\hat{\s}}$.
Moreover
$\RRR_{\s}\cap\RRR_{\hat{\s}}$ is a symmetric set with respect to the origin.
\end{cor}
In Theorem~\ref{thm:main}, we show that if the substitution satisfies some additional and natural hypotheses,
the set $\RRR_{\s}\cap\RRR_{\hat{\s}}$ has non-empty interior.
\subsection{Balanced pair algorithm}\label{sec:bpa}
In this section we introduce the balanced pair algorithm for two substitutions
$\sigma_1$ and $\sigma_2$ having the same incidence matrix.
We shall assume that the substitutions are primitive.
This algorithm was introduced in~\cite{sellami:1} and~\cite{sellami:2}, in the context of the study of
intersection of Rauzy fractals.
\smallskip
Let $U$ and $V$ be two finite words, we say that
$\begin{pmatrix}
U\\
V\\
\end{pmatrix}$
is a {\em balanced pair} if ${\bf l}(U) = {\bf l}(V)$,
where ${\bf l}(U)$ is the $k$-dimensional vector that gives the occurrences of the different symbols of the word $U$.
\smallskip
Given a word $U$ we denote by $\langle U\rangle_m$ the proper prefix of $U$ of length $m$.
A {\em minimal balanced pair} is a balanced pair
$\begin{pmatrix}
U\\
V\\
\end{pmatrix}$, such that
${\bf l}(\langle U\rangle_m) \neq {\bf l}(\langle V\rangle_m)$, for $1\leq m < |U|$.
\smallskip
Let $\sigma_1$ and $\sigma_2$ be two irreducible Pisot substitutions with the same incidence matrix.
Let $u$ and $v$ be the elements of $\A^{\N}$, which are fixed points of $\s_1$ and $\s_2$, respectively.
We define the balanced pair algorithm associated with the substitutions $\s_1$ and $\s_2$ as follows:
\smallskip
\noindent
We suppose that there exist prefixes $U$ and $V$ of $u$ and $v$, respectively, such that $\begin{pmatrix} U\\ V\end{pmatrix}$ is a minimal balanced pair.
We call this pair the first minimal balanced pair, of $u$ and $v$.
Under the right hypotheses, considered in section~\ref{sec:intersection}, the first minimal balanced pair always exists.
We apply the substitutions $\s_1$ and $\s_2$ to this balanced pair, in the following manner
$\begin{pmatrix} U\\ V\end{pmatrix}\rightarrow \begin{pmatrix} \sigma_1(U)\\ \sigma_2( V)\end{pmatrix}$.
Since the substitutions $\s_1$ and $\s_2$ have the same incidence matrix, the pair
$ \begin{pmatrix} \sigma_1(U)\\ \sigma_2( V)\end{pmatrix}$ is minimal.
We consider this new balanced pair and we decompose it into minimal balanced pairs.
We repeat this procedure to each of this new minimal balanced pairs.
Under the right hypotheses, considered in the next section,
the set of minimal balanced pairs is finite, and the algorithm terminates.
\section{Intersection of Rauzy fractals}\label{sec:intersection}
Let $\sigma_1$ and $\sigma_2$ be two unimodular irreducible Pisot substitutions with the same incidence matrix.
We consider their respective Rauzy fractals $\RRR_{\sigma_1}$ and $\RRR_{\sigma_2}$.
Since the origin is always an element of $\RRR_{\s_1}$ and $\RRR_{\s_2}$,
the intersection of $\RRR_{\sigma_1}$ and $\RRR_{\sigma_2}$ is non-empty, and it is a compact set because it is intersection of two compacts sets.
Let $\EEE$ be the closure of the intersection of the interior of $\RRR_{\sigma_1}$
and the interior of $\RRR_{\sigma_2}$.
Through out the article, we shall assume that $0$ is an interior point of one of the Rauzy fractals.
\begin{proposition}
Let $\sigma_1$ and $\sigma_2$ be two unimodular irreducible Pisot substitution with the same incidence matrix. We consider $\RRR_{\sigma_1}$ and $\RRR_{\sigma_2}$ their associated Rauzy fractal. We suppose that $0$ is an inner point to $\RRR_{\sigma_1}$. Then the set $\EEE$ has
non-empty interior and strictly positive Lebesgue measure.
\end{proposition}
\begin{proof}
By the assumption that $0$ is an inner point of $\RRR_{\sigma_1}$,
there exists an open set ${\mathcal U}$ such that $0\in {\mathcal U} \subset \RRR_1$.
Since the Rauzy fractal is the closure of its interior~(\cite{sirvent-wang})
and $0$ is a point of $\RRR_{\sigma_2}$, there
exists a sequence of points $\{x_n\}_{n\in\N}$ in the interior of
$\RRR_{\sigma_2}$ that converges to $0$.
Thus there exist open sets ${\mathcal V}_n$ such
that $x_n\in {\mathcal V}_n\subset \RRR_{\sigma_2}$.
From the fact $\{x_n\}$
converges to $0$, we conclude that, there exists $N\in\N$ such that $x_N\in {\mathcal U}$.
Therefore the open set ${\mathcal U}\cap {\mathcal V}_N$ is non-empty and ${\mathcal U}\cap {\mathcal V}_N\subset \RRR_{\sigma_1}\cap\RRR_{\sigma_2}$.
This implies that $\EEE$ contains a non-empty open
set; hence it has strictly positive Lebesgue measure.
\end{proof}
If the substitutions $\s_1$ and $\s_2$ satisfy the Pisot conjecture and $0$ is an inner point of one of the Rauzy fractals, then
the set $\EEE$ is also a Rauzy fractal associated with the substitution defined by the balanced pair algorithm.
This result was proved in~\cite{sellami:2} and we give here an idea of the proof.
\smallskip
\begin{thm}
\label{thm:nonemptyinterior}
Let $\sigma_1$ and $\sigma_2$ be two unimodular irreducible Pisot substitutions with the same incidence matrix.
Let $\RRR_{\sigma_1}$ and $\RRR_{\sigma_2}$ be their two
associated Rauzy fractals.
Suppose that $0$ is an inner point of $\RRR_{\sigma_1}$ and both substitutions satisfy the Pisot conjecture.
We denote by $\EEE$
the closure of the intersection of the interiors of $\RRR_{\sigma_1}$ and $\RRR_{\sigma_2}$.
Then $\EEE$ has non-empty
interior, and it is a Rauzy fractal associated with a Pisot substitution $\Sigma$ on the alphabet of minimal balanced pairs.
\end{thm}
We will assume the following lemma (for the proof see~\cite{sellami:2}), and we give an idea of the proof of Theorem~\ref{thm:nonemptyinterior}.
\begin{lemma}\label{lemma}
Let $\sigma_1$ and $\sigma_2$ be two unimodular irreducible Pisot substitutions
with the same incidence matrix. Let $\RRR_{\sigma_1}$ and $\RRR_{\sigma_2}$ be their
associated Rauzy fractals.
Suppose that ${\sigma_1}$ satisfies the Pisot conjecture and $0$ is an inner point of $\RRR_{\sigma_1}$.
Let $u$ and $v$ be the one-sided fixed points of $\sigma_1$ and $\sigma_2$ respectively. There exists a finite non-empty set $E$ of minimal balanced pairs, $E = \left\{\begin{pmatrix}U_1\\V_1\end{pmatrix},\ldots, \begin{pmatrix}U_p\\V_p\end{pmatrix}\right\}$, such that
the double sequence $\begin{pmatrix}u\\v \end{pmatrix}$ can be decomposed with elements from $E$.
\end{lemma}
The intersection set can be obtained as the projection of a fixed point of a new substitution
defined on the set of minimal balanced pairs.
Let $u$ and $v$ be the elements of $\A^{\N}$, such that $\s_1(u)=u$ and $\s_2(v)=v$.
From the hypotheses that $0$ is an inner point of $\RRR_{\s_1}$ and $\s_1$ satisfies the Pisot conjecture,
follows that there exist $W_1$ and $W_2$ prefixes of $u$ and $v$ respectively,
such that ${\bf l}(W_1)$ and ${\bf l}(W_2)$, i.e.,
the pair $\begin{pmatrix}W_1\\W_2\end{pmatrix}$ is a balanced pair, see Lemma 4.2 of~\cite{sellami:2}.
We decompose this balanced pair into minimal balanced pairs.
We repeat this procedure to each new minimal balanced pair.
By Lemma~\ref{lemma} the set of minimal balanced pair is finite.
This follows from the fact the set of common return times is bounded,
by iteration with $\sigma_1$ and $\sigma_2$, so
we obtain in bounded finite time the set of all minimal balanced pairs.
\smallskip
We take the image of each element of the finite set of minimal balanced pairs.
The substitution $\Sigma$ is defined as $\Sigma : \begin{pmatrix}U \\ V \end{pmatrix} \longmapsto \begin{pmatrix}\sigma_1(U)\\ \sigma_2(V)\end{pmatrix}$. The balanced pair $\begin{pmatrix}\sigma_1(U)\\ \sigma_2(V)\end{pmatrix}$ can be decomposed into minimal balanced pairs, and we can write the image of each minimal balanced pair with concatenated minimal balanced pairs.
So $\Sigma$ is a substitution defined on the set of minimal balanced pairs.
This substitution is Pisot, with the same dominant eigenvalue as $\s_1$~({\em cf.} Lemma 4.5 of~\cite{sellami:2}), however in general it is reducible.
Let $m$ be the number of different minimal balanced pairs, clearly $m\geq k$.
We consider the decomposition of the $m$-dimensional Euclidean space, in the corresponding expanding, contracting and complementary spaces:
$$
\R^m=E^s_0\oplus E^u_0 \oplus E^c_0.
$$
Let $E^s$ and $E^u$ be the contracting and expanding eigen-spaces corresponding to $\s_1$, since it is irreducible, we have $\R^k=E^u\oplus E^u$.
The substitutions $\Sigma$ and $\s$ have the same dominant eigenvalue, therefore
$E^s_0=E^s$ and $E^u_0=E^u$.
Let $\pi:\R^k\ra E^s$ be the projection of $\R^k$ onto $E^s$; and
$\pi_{\Sigma}:\R^m\ra E^s_0$ be the projection of $\R^m$ onto $E^s_0$.
If $\pi':\R^m\ra \R^k$ is the projection of $\R^m$ onto $\R^k$, then $\pi_{\Sigma}=\pi\circ\pi'$.
Let $(L'_n)_{n\in\N}$ be the broken line in $\R^m$ corresponding to a fixed point of the substitution $\Sigma$.
And let $(L_n)_{n\in\N}$ be the broken line in $\R^k$ corresponding to the fixed point of $\s_1$.
The points $\pi'(L'_n)$ corresponds to exactly the common points to the broken lines of $\s_1$ and $\s_2$~({\em cf.}~\cite{sellami:2}).
So for all $n\geq 0$, there exists $n'\in\N$, such that $\pi'(L'_n)=L_{n'}$, and moreover
if the point $L_l$ is a point common to the broken lines of $\s_1$ and $\s_2$, then there exists $n_l$ such that $\pi'(L_{n_l})=L_l$.
Hence
$$
\overline{\left\{\pi_{\Sigma}(L'_n)\,:\, n\in\N\right\}} = \RRR_{\s_1}\cap\RRR_{\s_2}.
$$
\smallskip
When we use Theorem~\ref{thm:nonemptyinterior} in the case of the substitutions $\s$ and $\hat{\s}$, we obtain the following result:
\begin{thm}\label{thm:main}
Let $\s$ be a unimodular irreducible Pisot substitution
and $\hat{\s}$ its reverse substitution.
Let $\RRR_{\s}$ and $\RRR_{\hat{\s}}$ be the respective Rauzy fractals.
We suppose that the substitution $\s$ satisfies the Pisot conjecture
and the origin is an inner point of $\RRR_{\s}$.
Then the set $\RRR_{\s}\cap\RRR_{\hat{\s}}$ has non-empty interior and is a Rauzy fractal
associated with the substitution obtained by balanced pair algorithm of $\s$ and $\hat{\s}$.
\end{thm}
\section{Examples}\label{sec:examples}
In this section we use letters to represent the elements of the alphabet $\A$.
\smallskip
\noindent
\textbf{Example 1:}\\
We consider the two substitutions $\s_1$ and $\s_2$ defined as:
\begin{center}
$\s_1:\left\{
\begin{array}{ll}
a\rightarrow aba\\
b\rightarrow ab
\end{array}\right.$
\hspace{1cm} and \hspace{1cm} $\s_2:\left\{
\begin{array}{ll}
a\rightarrow aba\\
b\rightarrow ba.
\end{array}\right.$
\end{center}
The Rauzy fractal of $\s_1$ is an interval, so by Corollary~\ref{cor:symmetricfractal} the Rauzy fractal of $\s_2$ is also an interval.
\smallskip
We describe the balanced pair algorithm and we obtain a morphism that characterize the common points of these two Rauzy fractals.
In this example, the first minimal balanced pair that we can consider is the beginning of the two fixed points associated with $\s_1$ and $\s_2$ it will be
$\begin{pmatrix}
a\\
a\\
\end{pmatrix}$.
We represent the image of the first element of this pair by $\s_1$ and
the second one by $\s_2$. We obtain :
$\begin{pmatrix}
a\\
a\\
\end{pmatrix}\overset{\s_1, \s_2}\longrightarrow\begin{pmatrix}
aba\\
aba\\
\end{pmatrix}$.
We denote by $A$ the minimal balanced pair $\begin{pmatrix}
a\\
a\\
\end{pmatrix}$ and by $B$ the minimal balanced pair $\begin{pmatrix}
b\\
b\\
\end{pmatrix}$.
Hence we obtain $A\rightarrow ABA.$
\smallskip
The second step is to consider the same process with the new balanced pair
$B = \begin{pmatrix}
b\\
b\\
\end{pmatrix}$.
We consider the image of this balanced pair with the two substitutions $\s_1$ and
$\s_2$, and we obtain:
\begin{center}
$\begin{pmatrix}
b\\
b\\
\end{pmatrix}\overset{\s_1, \s_2}\longrightarrow\begin{pmatrix}
ab\\
ba\\
\end{pmatrix}$.\\
\end{center}
We obtain an other balanced pair $\begin{pmatrix}
ab\\
ba\\
\end{pmatrix}$
and we denote by $C$ this new balanced pair. We get the image
of $B$ which is $C$.
We continue with this algorithm and we obtain the image of the balanced pair
$\begin{pmatrix}
ab\\
ba\\
\end{pmatrix}$ is the new balanced pair $\begin{pmatrix}
abaab\\
baaba\\
\end{pmatrix}$.
Therefore we obtain that the image of the letter $C$ is $CAC$.
On total, we obtain an alphabet $\mathcal{B}$ on $3$ symbols and we can define the morphism $\Sigma$ as :
\begin{center}
$\Sigma :\left\{
\begin{array}{ll}
A\rightarrow ABA,\\
B\rightarrow C,\\
C\rightarrow CAC.\\
\end{array}\right. $
\end{center}
This morphism $\Sigma$ is the substitution obtained in Theorem~\ref{thm:nonemptyinterior}.
The characteristic polynomial of the transition matrix of $\Sigma$ is $(x^2-3x+1)(x-1)$.
The substitution $\Sigma$ generates all the common points of the two Rauzy fractals
associated with $\s_1$ and $\s_2$.
\medskip
\noindent
\textbf{Example 2:}\\
In this example we consider the family of Pisot substitutions defined as follows:
\begin{center}
$\sigma_{1,i}:\left\{
\begin{array}{ll}
a\rightarrow a^ib,\\
b\rightarrow a^ic,\\
c\rightarrow a,
\end{array}\right.$
\hspace{1cm} and \hspace{1cm} $\sigma_{2,i}:\left\{
\begin{array}{ll}
a\rightarrow ba^i,\\
b\rightarrow ca^i,\\
c\rightarrow a.
\end{array}\right.$
\end{center}
\begin{figure}[h]
\begin{center}
\scalebox{0.3}{\includegraphics{23.eps}}
\vspace{0.5cm}
\scalebox{0.3}{\includegraphics{13.eps}}
\caption{ Rauzy fractals associated with $\sigma_{1,3}$, $\sigma_{2, 3}$. \label{fig:ex2-fractals}}
\end{center}
\end{figure}
Some geometrical and dynamical properties of
the Rauzy fractals of this family of substitutions have been studied in~\cite{LMST,thuswaldner}.
In particular
the symmetry of these Rauzy fractals been studied in~\cite{sirvent:2},
further properties of these Rauzy fractals were studied in~\cite{NSS}.
They are symmetric, but their center of symmetry is not the origin.
The Rauzy fractals for $\s_{1,3}$ and $\s_{2,3}$ are shown in Figure~\ref{fig:ex2-fractals}.
The classical tribonacci substitution, is $\s_{1,1}$.
The Rauzy fractals of $\s_{1,1}$, $\s_{2,1}$ and their intersections are shown in Figure~\ref{fig:tribo}.
Since $\s_{2,i}$ is the reverse substitution of $\s_{1,i}$, both substitutions have the same incidence matrix:
$$M_i = \begin{pmatrix}
i & i & 1 \\
1 & 0 &0 \\
0 & 1 & 0 \\
\end{pmatrix}.
$$
Let $P_i(x) = x^3 - ix^2- ix-1$ be the characteristic polynomial of $M_i$. The substitutions $\sigma_{1,i}$ and $\sigma_{2,i}$ are unimodular irreducible Pisot substitutions,~({\em cf.}~\cite{brauer}).
We are interested in this section to study the substitution associated with the intersection.
In the following proposition we prove that intersection substitution is defined in an alphabet of six symbols for all $i\geq 1$.
\begin{proposition}
The intersection substitution $\Sigma_i$ associated with $\sigma_{1,i}$ and $\sigma_{2,i}$ is defined on an alphabet of six symbols as follows:
\begin{center}
$\Sigma_i:\left\{
\begin{array}{ll}
A\rightarrow B,\\
B\rightarrow C,\\
C\rightarrow [(AD)^iAE]^i(AD)^iA,\\
D\rightarrow F,\\
E\rightarrow (AD)^{i-1}A,\\
F\rightarrow [(AD)^iAE]^{i-1}(AD)^iA.
\end{array}\right.$
\end{center}
\end{proposition}
\begin{proof}
We apply the balanced pair algorithm to $\sigma_{1,i}$ and $\sigma_{2,i}$. The first minimal balanced pair is $A = \begin{pmatrix}
a\\
a\\
\end{pmatrix}$. We take the image of $A$ with $\sigma_{1,i}$ and $\sigma_{2,i}$ we obtain a new balanced pair $B = \begin{pmatrix}
a^i b\\
ba^i\\
\end{pmatrix}$. The balanced pair $B$ is a minimal balanced pair. So we obtain $A\longrightarrow B$. We take its image again, we obtain:
$$\Biggl(\begin{matrix}
a^ib\\
ba^i\\
\end{matrix}\Biggr)\overset{\sigma_{1,i}, \sigma_{2,i}}\longrightarrow \Biggl(\begin{matrix}
(a^ib)^ia^ic\\
ca^i(ba^i)^i\\
\end{matrix}\Biggr).$$
\smallskip
We denote by $C = \Biggl(\begin{matrix}
(a^ib)^ia^ic\\
ca^i(ba^i)^i\\
\end{matrix}\Biggr)$, the new balanced pair, it is clear that $C$ is a minimal balanced pair and $B\longrightarrow C$. We continues with the algorithm we calculate the image of $C$, we obtain:
$$\Biggl(\begin{matrix}
(a^ib)^ia^ic\\
ca^i(ba^i)^i\\
\end{matrix}\Biggr) \overset{\sigma_{1,i}, \sigma_{2,i}}\longrightarrow \Biggl
(\begin{matrix}
[(a^ib)^ia^ic]^i(a^ib)^ia\\
a(ba^i)^i[ca^i(ba^i)^i]^i\\
\end{matrix}\Biggr).
$$
Note that the right hand side term can be written as
$$
\Biggl(\begin{matrix}
[(a^ib) \ldots (a^ib) a^ic]\ldots [(a^ib) \ldots (a^ib) a^ic](a^ib) \ldots (a^ib)a \\
a(ba^)i\ldots (ba^i)[ca^i(ba^i)\ldots (ba^i)]\ldots [ca^i(ba^i)\ldots (ba^i)]\\
\end{matrix}\Biggr). $$
We can decompose the new balanced pair as follows:
$$\Biggl[\Biggl(\begin{matrix}
a\\
a\\
\end{matrix}\Biggr) \Biggl(\begin{matrix}
a^{i-1}b\\
ba^{i-1}\\
\end{matrix}\Biggr)\ldots \Biggl(\begin{matrix}
a\\
a\\
\end{matrix}\Biggr) \Biggl(\begin{matrix}
a^{i-1}b\\
ba^{i-1}\\
\end{matrix}\Biggr) \Biggl(\begin{matrix}
a\\
a\\
\end{matrix}\Biggr) \Biggl(\begin{matrix}
a^{i-1}c\\
ca^{i-1}\\
\end{matrix}\Biggr)\Biggr]^i \Biggl[\Biggl(\begin{matrix}
a\\
a\\
\end{matrix}\Biggr) \Biggl(\begin{matrix}
a^{i-1}b\\
ba^{i-1}\\
\end{matrix}\Biggr)\Biggr]^i.$$
So we obtain two new minimals balanced pairs, we denote $D = \Biggl(\begin{matrix}
a^{i-1}b\\
ba^{i-1}\\
\end{matrix}\Biggr)$ and $E = \Biggl(\begin{matrix}
a^{i-1}c\\
ca^{i-1}\\
\end{matrix}\Biggr).$ The image of C is $ [(AD)^iAE]^i(AD)^iA$. We applies the balanced pair algorithm to $D$ we obtain a new balanced pair $F = \Biggl(\begin{matrix}
(a^ib)^{i-1}a^ic\\
ca^i(ba^i)^{i-1}\\
\end{matrix}\Biggr)$. Again $F$ is a minimal balanced pair.
We continue with minimal balanced pair $E$, we obtain $E\longrightarrow (AD)^iA$ and finally $F\longrightarrow [(AD)^iAE]^i(AD)^iA$.
\end{proof}
\begin{figure}
\begin{center}
\scalebox{0.25}{\includegraphics{222.eps}}
\vspace{0.2cm}
\scalebox{0.25}{\includegraphics{333.eps}}
\caption{\label{fig:intersection-ex2} Rauzy fractals intersection $\Sigma_{2}$ and $\Sigma_{3}$. }
\end{center}
\end{figure}
The characteristic polynomial of $\Sigma$ is
$$P_{\Sigma_i}(x) = (x^3 - ix^2- ix-1)(x^3+ix^2+ix-1).$$
Figures~\ref{fig:intersection-ex2} and~\ref{fig:tribo} show the intersection sets for the first three substitutions of this family.\\
\begin{figure}
\begin{center}
\scalebox{0.2}{\includegraphics{trib2.eps}}
\hspace{0.2cm}
\scalebox{0.2}{\includegraphics{trib1.eps}}
\vspace{0.25cm}
\scalebox{0.2}{\includegraphics{111.eps}}
\caption{\label{fig:tribo} Rauzy fractals of $\sigma_{1,1}$, $\sigma_{2,1}$ and $\Sigma_1$. }
\end{center}
\end{figure}
\medskip
\noindent
\textbf{Example 3:}\\
In this example we consider the two substitutions defined as follows:
\begin{center}
$\sigma_1:\left\{
\begin{array}{ll}
a\rightarrow ab,\\
b\rightarrow ca,\\
c\rightarrow a,
\end{array}\right.$
\hspace{1cm} and \hspace{1cm} $\sigma_2:\left\{
\begin{array}{ll}
a\rightarrow ba,\\
b\rightarrow ac,\\
c\rightarrow a.
\end{array}\right.$
\end{center}
The substitution $\s_1$ is known as the flipped tribonacci substitution~(\cite{sirvent:1}).
When we apply the balanced pair algorithm to these two substitution,
we obtain the substitution $\Sigma$ for intersection on $15$ symbols defined as:
$$
\begin{array}{lllll}
A\rightarrow B, & B\rightarrow ACA, & C\rightarrow D, & D\rightarrow E, & E\rightarrow AFA,\\
F\rightarrow DGHGD, & G\rightarrow I, & H\rightarrow JKJ, & I\rightarrow J, & J\rightarrow ALA,\\
K\rightarrow AMAMA, & L\rightarrow DGD, & M\rightarrow N, & N\rightarrow AOA, & O\rightarrow AMAMAMA.
\end{array}
$$
The characteristic polynomial of the transition matrix of $\Sigma$ is
$$
(x - 1) (x + 1) (x^2 - x + 1) (x^3 - x^2 - x - 1)
(x^3 + x^2 + x- 1) (x^5 + x^4 - 2x^2 - 3x + 1).
$$
The Rauzy fractal associated with $\Sigma$, i.e., the intersection of the Rauzy fractals of $\s_1$ and $\s_2$ is shown in Figure~\ref{fig:ex3}.
\begin{figure}
\begin{center}
\scalebox{0.2}{\includegraphics{baaca.eps}}
\hspace{0.2cm}
\scalebox{0.2}{\includegraphics{abcaa.eps}}
\vspace{0.25cm}
\scalebox{0.3}{\includegraphics{aaa111.eps}}
\caption{\label{fig:ex3} Rauzy fractals of Example 3 and their intersection. }
\end{center}
\end{figure}
\medskip
\noindent
\textbf{Example 4:}\\
In the previous examples, we have seen that the substitution $\Sigma$ has the property: $\Sigma(U)$ is a palindrome for each symbol $U$ in the alphabet where $\Sigma$ is defined.
Here we present an example in which this situation does not occur.
Consider the substitutions in two symbols, given by
\begin{center}
$\s_1:\left\{
\begin{array}{ll}
a\rightarrow aabbaabab\\
b\rightarrow ab
\end{array}\right.$
\hspace{1cm} and \hspace{1cm} $\s_2:\left\{
\begin{array}{ll}
a\rightarrow babaabbaa\\
b\rightarrow ba.
\end{array}\right.$
\end{center}
We remark that these substitutions are unimodular irreducible Pisot.
When we run the balanced pair algorithm, we get the following balanced pairs:
$$
\Biggl(\begin{matrix}
aabb\\
baba\\
\end{matrix}\Biggr),
\Biggl(\begin{matrix}
abab\\
bbaa\\
\end{matrix}\Biggr),
\Biggl(\begin{matrix}
aab\\
baa\\
\end{matrix}\Biggr),
\Biggl(\begin{matrix}
abaabb\\
bbaaba\\
\end{matrix}\Biggr),
\Biggl(\begin{matrix}
aabab\\
babaa\\
\end{matrix}\Biggr).
$$
If we denote them by $A, B,C,D, E$, respectively.
The resulting substitution on this alphabet is:
$$
\Sigma:\left\{
\begin{array}{l}
A\ra ACDEB,\\
B\ra AEDCB,\\
C\ra ACDCB,\\
D\ra AEDCDEB,\\
E\ra ACDEDCB.
\end{array}
\right.
$$
It can be observed that $\Sigma(U)$ is not a palindrome, for any $U$ in the alphabet.
The characteristic polynomial of the matrix associated with $\Sigma$ is
$$
x^2(x-1)(x^2-6x+1).
$$
\medskip
\noindent
\textbf{Example 5:}\\
Let $\s$ be the substitution on three symbols defined by
$$
\begin{array}{ccc}
a\ra abc, & b\ra a, & c\ra ac.
\end{array}
$$
It has a unique one-sided fixed point:
$$u:=\text{``}\s^{\infty}(a)\text{"}=abcaacabcabcacabcaacabca\cdots.$$
We conjecture the origin is a boundary point of its Rauzy fractal.
Let $\hat{\s}$ be its reversed substitution. It has a unique one-sided fixed point:
$$
v:=\text{``}\hat{\s}^{\infty}(c)\text{"}=cacbcaacbacacbacbacaacba\ldots.
$$
We conjecture that there is no initial balanced pair between $u$ and $v$, i.e.
the balanced pair algorithm it cannot be applied to $u$ and $v$.
If the conjecture is true, then the intersection of both Rauzy fractals is reduced to the origin.
Figure~\ref{fig:ex5} shows the Rauzy fractals of $\s$ and $\hat{\s}$.
\begin{figure}
\begin{center}
\scalebox{0.2}{\includegraphics{000.eps}}
\hspace{0.2cm}
\scalebox{0.2}{\includegraphics{001.eps}}
\caption{\label{fig:ex5} Rauzy fractals of Example 5.}
\end{center}
\end{figure}
\section{Remarks and open questions}\label{sec:remarks}
\begin{enumerate}
\item
We say that a substitution $\s$ satisfies the strong coincidence condition on prefixes (respectively on suffixes) if for any two symbols $i,j\in\A$ then there exists $n\in\N$, $a\in\A$ and $p,q,r,t\in\A^*$, such that
$$
\begin{array}{lr}
\s^n(i)=pat \text{ and } \s^n(j)=qar, & \text{ with } {\bf l}(p)={\bf l}(q)
\end{array}
$$
$$
\text{ (respectively } {\bf l}(t)={\bf l}(r) \text{).}
$$
Every irreducible unimodular Pisot substitution in two symbols satisfies the strong coincidence condition~({\em cf.}~\cite{barge-diamond}).
It is conjectured that all irreducible unimodular Pisot substitutions satisfies the strong coincidence condition.
Clearly a substitution $\s$ satisfies the strong coincidence condition on prefixes (or suffixes) if and only if $\hat{\s}$ satisfies the strong coincidence condition on suffixes (or prefixes).
If $\s^n$ has a unique fixed point in $\A^{\N}$ for all $n\geq 1$ then it satisfies the strong coincidence condition.
\item The Rauzy fractal $\RRR_{\s}$ admits a partition
$\{\RRR_{\s}(1),\ldots,\RRR_{\s}(k)\}$, where
$$
\RRR_{\s}(j):=\overline{\left\{\pi(L_n)\,|\, u_n=j \text{ and } n\in\N \right\}}.
$$
This partition is called the {\em natural decomposition} of $\RRR_{\s}$.
The set $\RRR_{\s}$ and its natural decomposition can be obtained as
a fixed point of a graph directed iterated function system, for details see~\cite{sirvent-wang}.
If the substitution $\s$ satisfies the strong coincidence condition,
then the sets
$\RRR_{\s}(j)$ are measure-wise disjoint~(\cite{arnoux-ito}).
We note that the natural decomposition of $\RRR_{\hat{\s}}$ does not have to be
the reflection through the origin of the natural decomposition of $\RRR_{\s}$,
as it can be seen in Figures~\ref{fig:ex2-fractals} and~\ref{fig:tribo},
for some substitutions considered in Example 2.
\item Is it possible to generalize the construction described in this article in the reducible and/or non-unimodular case?
\item We wonder if it is possible to obtain subsets of Rauzy fractals with other symmetries, via the balanced pair algorithm.
\item Is the Hausdorff dimension of the boundary of the intersection set the same of the dimension of the boundary of the Rauzy fractal?
\item Let $p(x)$ be the characteristic polynomial of the matrix $M_{\s}$ and $q(x)$ its reciprocal.
In examples 2 and 3, $p(x)$ and $q(x)$ are factors of
the characteristic polynomial of the matrix $M_{\Sigma}$,
where $\Sigma$ is the substitution obtained by the balanced pair algorithm of $\s$ and $\hat{\s}$.
In examples 1 and 4 we have that $p(x)=q(x)$.
We conjecture that if $p(x)\neq q(x)$ then $p(x)$ and $q(x)$ are factors of
the characteristic polynomial of the matrix $M_{\Sigma}$, when $\s$ is unimodular irreducible Pisot.
\end{enumerate}
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TITLE: Smooth morphism of smooth varieties with fibres isomorphic to an affine space
QUESTION [7 upvotes]: Let $X$ and $Y$ be smooth varieties over the field of complex numbers $\bf C$
(smooth integral separated schemes of finite type over $\bf C$). Let
$$f\colon X\to Y$$
be a surjective morphism such that
for any closed point $y\in Y$, the schematic fibre $f^{-1}(y)\subset X$
is isomorphic to the affine space ${\Bbb A}_{\bf C}^{n(y)}$.
Moreover, assume that the morphism $f$ is smooth (which is equivalent to the assumption that $n(y)$
is the constant function $n(y)=n$, where $n=\dim X-\dim Y$).
Consider the real $C^\infty$-manifolds $X^\infty=X({\bf C})$ and $Y^\infty=Y({\bf C})$ and the induced $C^\infty$-map
$$f^\infty\colon X^\infty\to Y^\infty.$$
Since $f$ is smooth, the map $f^\infty$ is a submersion, that is, for any $x\in X^\infty$, the differential
$$d_x f\colon T_x(X)\to T_{f(x)}Y$$
is surjective. Moreover, each fibre of $f^\infty$ is diffeomorphic to ${\bf R}^{2n}$.
By Corollary 31 of G. Meigniez, Submersions, fibrations and bundles,
Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771-3787,
the map $f^\infty$ is a locally trivial fibre bundle of $C^\infty$-manifolds, that is, for any $y\in Y^\infty$
there exists an open neighborhood ${\mathcal U}_y$ of $y$ in $Y^\infty$
such that $f^{-1}({\mathcal U}_y)\simeq {\bf R}^{2n}\times {\mathcal U}_y$,
where $\simeq$ denotes a $C^\infty$-diffeomorphism compatible with the projections onto ${\mathcal U}_y$.
Question 1. Does it follow that the morphism $f$
is a locally trivial fibre bundle in the étale topology,
that is, for any closed point $y\in Y$
there exists an étale open neighborhood $ U_y\to Y$ of $y$ such that
$$X\times_Y U_y\simeq {\Bbb A}_{\bf C}^n\times_{\bf C} U_y\,,$$
where $\simeq$ denotes an isomorphism of $\bf C$-varieties
compatible with the projections onto $U_y$ ?
Question 2. Is $f$ a locally trivial fibre bundle in the flat topology?
Consider the complex analytic manifolds $X^{\rm an}=X({\bf C})$, $Y^{\rm an}=Y({\bf C})$
and the induced complex analytic morphism
$$f^{\rm an}\colon X^{\rm an}\to Y^{\rm an}.$$
Question 3. Is $f^{\rm an}\colon X^{\rm an}\to Y^{\rm an}$
a locally trivial fibre bundle of complex analytic manifolds,
that is, for any $y\in Y^{\rm an}$
there exists an open neighborhood ${\mathcal U}_y$ of $y$ in $Y^{\rm an}$
such that $(f^{\rm an})^{-1}({\mathcal U}_y)\simeq {\bf C}^n\times {\mathcal U}_y$,
where $\simeq$ denotes an analytic isomorphism compatible with the projections onto ${\mathcal U}_y$ ?
REPLY [3 votes]: Regarding Question 1, it seems to be an open problem, known as a variant of Dolgachev–Weisfeiler Conjecture. The article $\mathbb{A}^2$-fibrations between affine spaces are $\mathbb{A}^2$-trivial (A. Dubouloz) shows that an $\mathbb{A}^2$-fibration $f\colon X\to S$ is étale-locally trivial if and only if $\Omega^1_{X/S}$ is a pullback of a locally-free sheaf $\mathcal{E}$ on $S$. Similar questions are also mentioned in Vénéreau polynomials and related fiber bundles (S. Kaliman, M. Zaidenberg), page 276. Perhaps some experts can answer this question in greater detail.
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Home » ICT » Infrastructure/Network Security » IIN-3971116
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A rack server also called as a rack mount server is a type of hardware which is installed in a certain framework which is downright horizontal instead of upright tower server system. This enables the servers to execute and manage an enterprise application or serve as data center. A single rack comprises of several servers stacked over one another making a joint network resource which minimizes the floor space. The global data center rack market is anticipated to grow at the CAGR of 11% during the forecast period 2016-2023. The market generated the revenue of $1.6 Billion in 2015 and is anticipated to reach up to $3.32 Billion by 2023. The key factors driving the market include:
The increase in demand for high density servers is due to the benefits associated with it such as cost effectiveness, minimum respond time, and long term savings.
Expansion of infrastructural market, infrastructural upgradation in data center, high investment and high density server environment helps to drive the global data center rack market. The increasing internet of things market helps to accelerate the global data center rack market. It is projected that the amount of joined devices globally will rise from $15 billion in 2016 to 50 billion by 2020. It is estimated that the expenditure on internet of things devices and services will rise from $656 billion in 2014 to $1.7 trillion in 2020. However, some of the major factors that restraint the growth of global data center rack market are reduction of shipments of rack, decline in demand of server hardware due to consolidation and virtualization and interoperability issues. Blade servers, micro servers and HVAC for highly dense environment are some of the challenges that the global data center rack market faces. In spite of restraining factors and challenges, the global data center rack market still has opportunities such as increasing trend of customized racks and need for seismic cabinets for critical environment.
Source: OBRC Analysis
The global data center rack market is segmented into rack units, frame design, frame size, services, vertical and end-users. Rack units are further segmented into:
Frame design is sub-segmented into:
Frame size is sub-segmented into:
Service is sub-segmented into:
Vertical is sub-segmented into:
End-user is sub-segmented into:
The revenue for the above verticals and end-user segments are specific to the data center rack market. However, the total revenue of these verticals and end-users in general has been excluded from the scope of the report. Moreover, the total market has been calculated by summing up rack unit, frame size, frame design and service segments and excluded other than these segments for calculating data center rack market revenue share.
Geographically, the global data center rack market is segmented into:
North America holds the largest share followed by Europe. Increasing popularity of high density data centers and the need to decrease the power consumption of data center infrastructure are major factors contributing to the growth of data center rack market in North America. Asia Pacific has the fastest growing rate during the forecast period, due to high technological adoption rates and rising number of new data center.
The major market players of the global DATA CENTER rack market are:
Detailed analysis of these companies provided in this report comprises of overview, SCOT analysis, product portfolio, strategic initiative and strategic analysis.
These companies use various strategies such as mergers & acquisitions, collaboration, partnership and product launching. For example, in May 2017, Cisco acquired Viptela a software company to increase the product portfolio of the company. Eaton launched new solutions and innovative products applicable to data centers to offer the solutions for making businesses smoother with low cost.
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\begin{document}
\title{Asymptotic Hodge Theory and Quantum Products}
\bibliographystyle{amsplain}
\address{Department of Mathematics and Statistics\\ University of
Massachusetts\\ Amherst\\ MA 01003}
\author{Eduardo Cattani}
\author{Javier Fernandez}
\email{cattani@math.umass.edu}
\email{jfernand@math.umass.edu}
\begin{abstract}
Assuming suitable convergence properties for the Gromov-Witten
potential of a Calabi-Yau manifold $X$, one may construct a
polarized variation of Hodge structure over the complexified
K\"ahler cone of $X$.
In this paper we show that, in the case of
fourfolds, there is a correspondence between ``quantum potentials''
and polarized variations of Hodge structures that degenerate to a
maximally unipotent boundary point. Under this correspondence, the
WDVV equations are seen to be equivalent to the Griffiths' trasversality
property of a variation of Hodge structure.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:introduction}
The Gromov-Witten potential of a Calabi-Yau manifold is a generating
function for some of its enumerative data. It may be written as
$\Phi^{GW} = \Phi_{0} + \Phi_{\hbar}$, where $\Phi_{0}$ is determined
by the cup product structure in cohomology. For a quintic
hypersurface $X\subset P^4$, the quantum potential $\Phi_{\hbar}$ is
holomorphic in a neighborhood of $0\in \C$; the coefficients of its
expansion
\begin{equation*}
\Phi_{\hbar}(q) \ :=\ \sum_{d=1}^\infty \langle
I_{0,0,d}\rangle\cdot q^d,
\end{equation*}
are the Gromov-Witten invariants $\langle I_{0,0,d}\rangle$ which
encode information about the number of rational curves of degree $d$
in $X$. This potential gives rise to a flat connection on the trivial
bundle over the punctured disk $\Delta^*$ with fiber $V = \oplus_p
H^{p,p}(X)$. This flat bundle is shown to underlie a polarized
variation of Hodge structure whose degeneration at the origin is, in
an appropriate sense, maximal. The Mirror Theorem in the context of
\cite{ar:CDGP-pair}, \cite{ar:LLY-mirror-1},
\cite{ar:givental-equivariant}, \cite[Theorem 11.1.1]{bo:CK-mirror}
asserts that this is the variation of Hodge structure arising from a
family of mirror Calabi-Yau threefolds and, therefore, that the
Gromov-Witten potential may be computed from the period map of this
family, written with respect to a canonical coordinate at
``infinity''. This leads to the effective computation (and
prediction) of the number of rational curves, of a given degree, in a
quintic threefold.
The Mirror Theorem, in the sense sketched above, has a conjectural
generalization for toric Calabi-Yau threefolds \cite[Conjecture
8.6.10]{bo:CK-mirror} but the situation in the higher dimensional case
is considerably murkier. Still, one may write a formal potential in
terms of axiomatically defined Gromov-Witten invariants and, assuming
suitable convergence properties, construct from it an abstract
polarized variation of Hodge structure. The third derivatives of
$\Phi^{GW}$ may be used to define a quantum product on $H^*(X)$ whose
associativity is equivalent to a system of partial differential
equations satisfied by $\Phi^{GW}$. These are the so-called WDVV
equations, after E.~Witten, R.~Dijkgraaf, H.~Verlinde, and
E.~Verlinde.
The purpose of this paper is to further explore the relationship
between quantum potentials and variations of Hodge structure. The
weight-three case has been extensively considered in
\cite{bo:CK-mirror, ar:greg-higgs}; here we consider the case of
structures of weight four.
In \S \ref{sec:variations_at_infinity}, after recalling some basic
notions of Hodge theory, we describe the behavior of variations at
``infinity'' in terms of two ingredients: a nilpotent orbit and a
holomorphic function $\Gamma$ with values in a graded nilpotent Lie
algebra. Nilpotent orbits may be characterized in terms of polarized
mixed Hodge structures; in the particular case when these split over
$\R$, their structure mimics that of the cohomology of a K\"ahler
manifold under multiplication by elements in the K\"ahler cone. For a
Calabi-Yau fourfold $X$, this action ---together with the intersection
form--- characterizes the cup product in $\oplus_p H^{p,p}(X)$. The
holomorphic function $\Gamma$, in turn, is completely determined by
one of its components, $\Gamma_{-1}$, relative to the Lie algebra
grading. Moreover, it must satisfy the differential equation
(\ref{eq:integcond}) involving the monodromy of the variation. This
is the content of Theorem~\ref{th:improved_2.8} which generalizes a
result of P.~Deligne \cite[Theorem 11]{ar:Del-local_behavior} for a
case when the variation of Hodge structure is also a variation of
mixed Hodge structure. Throughout this section, and indeed this whole
paper, we restrict ourselves to real variations of Hodge structure
without any reference to integral structures.
The asymptotic data associated with a variation of Hodge structure
depends on the choice of local coordinates. In
\S \ref{sec:canonical_coordinates} we describe how the nilpotent orbit
and $\Gamma$ behave under a change of coordinates and show that in
certain cases there are canonical choices of coordinates. This
happens, for example, in the cases of interest in mirror symmetry and
we recover, in this manner, Deligne's Hodge theoretic description of
these canonical coordinates.
In \S \ref{sec:frobenius} we begin our discussion of quantum products
by concentrating on their ``constant'' part. We define the notion of
polarized, graded Frobenius algebras and show that they give rise to
nilpotent orbits which are maximally degenerate in an appropriate
sense. Moreover, in the weight-four case this correspondence may be
reversed.
Finally, in \S \ref{sec:quantum} we define a notion of quantum
potentials in polarized, graded Frobenius algebras of weight four
that abstracts the main properties of the Gromov-Witten potential for
Calabi-Yau fourfolds. Following the arguments of \cite[\S
8.5.4]{bo:CK-mirror} we construct a variation of Hodge structure
associated with such a potential and explicitly describe its
asymptotic data. We show, in particular, that the function $\Gamma$
is completely determined by the potential in a manner that transforms
the WDVV equations into the horizontality equation
(\ref{eq:integcond}). This is done in Theorem~\ref{main} which
establishes an equivalence between such potentials and variations with
certain prescribed limiting behavior. For the case of weight three a
similar theorem is due to G.~Pearlstein \cite[Theorem
7.20]{ar:greg-higgs}.
A distinguishing property of the weight $3$ and $4$ cases is that the
quantum product on the $(p,p)$-cohomology is determined by the
$H^{1,1}$-module structure. In the case of constant products, this
statement is the content of
Proposition~\ref{prop:constant_prod_from_nilpotent}. For general
weights the PVHS on the complexified K\"ahler cone determines only the
structure of the $(p,p)$-cohomology as a $\sym H^{1,1}$-module,
relative to the small quantum product. At the abstract level, one can
prove that there is a correspondence between germs of PVHS of weight
$k$ at a maximally unipotent boundary point whose limiting mixed Hodge
structure is Hodge-Tate and flat, real families of polarized graded
Frobenius $\sym H^{1,1}$-modules over $\oplus_{p=0}^k H^{p,p}$.
Details will appear elsewhere.
Barannikov (\cite{ar:barannikov-1},~\cite{ar:barannikov-2} and
\cite{ar:barannikov-3}) has shown that for projective complete
intersections, the PVHS constructed from the Gromov-Witten potential
is of geometric origin and coincides with the variation arising from
the mirror family. He has introduced, moreover, the notion of
semi-infinite variations of Hodge structure. These more general
variations are shown to correspond to solutions of the WDVV equations.
\smallskip
\noindent{\bf Acknowledgments:} We are very grateful to Emma Previato
for organizing the Special Session on Enumerative Geometry in Physics
at the Regional AMS meeting of April 2000 in Lowell, Massachusetts and
for putting together this volume.
We also wish to thank David Cox for very helpful conversations and to
Serguei Barannikov for his useful comments.
\section{Variations at Infinity}
\label{sec:variations_at_infinity}
We begin by reviewing some basic results about the asymptotic behavior
of variations of Hodge structure. We refer to \cite{ar:CK-luminy,
ar:Gri-periods-1, bo:griffiths-topics, ar:S-vhs} for details and
proofs.
Let $M$ be a connected complex manifold, a (real) {\em variation of
Hodge structure} (VHS) $\VV$ over $M$ consists of a holomorphic
vector bundle $\VV\to M$, endowed with a flat connection $\nabla$, a
flat real form $\VV_{\R}$, and a finite decreasing filtration $\FF$ of
$\VV$ by holomorphic subbundles ---the \textsl{ Hodge filtration\/}---
satisfying
\begin{eqnarray}\label{horizontality}
\nabla\FF^p & \subset & \Omega^1_M
\otimes \FF^{p-1}\quad\hbox{(Griffiths' horizontality) and}
\\ \label{opposed}
\VV& = & \FF^p \oplus \conj{\FF}^{k-p+1}
\end{eqnarray}
for some integer $k$ ---the \textsl{weight\/} of the variation--- and
where $\conj{\FF}$ denotes conjugation relative to $\VV_{\R}$. As a
$C^{\infty}$-bundle, $\VV$ may then be written as a direct sum
\begin{equation}\label{bundledecomposition}
\VV = \bigoplus_{p+q=k}\ \VV^{p,q}\ ,\quad \quad \VV^{p,q} = \FF^p
\cap \conj{\FF}^q\,;
\end{equation}
the integers $h^{p,q} = \dim\,\VV^{p,q}$ are the \textsl{ Hodge
numbers\/}. A \textsl{ polarization\/} of the VHS is a flat
non-degenerate bilinear form $\ScS$ on $\VV$, defined over $\R$, of
parity $(-1)^k$, whose associated flat Hermitian form
$\ScS^h(\,.\,,\,.\,) = i^{-k}\, \ScS(\,.\,,\,\bar.\,)$ is such that
the decomposition~(\ref{bundledecomposition}) is $\ScS^h$-orthogonal
and $(-1)^p\ScS^h$ is positive definite on $\VV^{p,k-p}$.
Specialization to a fiber defines the notion of \textsl{polarized
Hodge structure} on a $\C$-vector space $V$. We will denote by $D$
\cite{ar:Gri-periods-1} the \textsl{ classifying space\/} of all
polarized Hodge structures of given weight and Hodge numbers on a
fixed vector space $V$, endowed with a fixed real structure $V_{\R}$
and the polarizing form $S$. Its Zariski closure $\check D$ in the
appropriate variety of flags consists of all filtrations $F$ in $V$,
with $\dim\,F^p = \sum_{r\geq p}\,h^{r,k-r}$, satisfying
$\,S(F^p,F^{k-p+1}) = 0\, $. The complex Lie group $G_{\C}=
\aut(V,S)$ acts transitively on $\check D$ ---therefore $\check D$ is
smooth--- and the group of real points $G_{\R}$ has $D$ as an open
dense orbit. We denote by $\jlg \subset \gl(V)$, the Lie algebra of
$G_{\C}$, and by $\lgr \subset \jlg$ that of $G_{\R}$. The choice
of a base point $F\in \check D$ defines a filtration in $\jlg$
\begin{equation*}
F^a\jlg = \{\,T\in\jlg\ :\ T\,F^p \subset F^{p+a}\,\}\,.
\end{equation*}
The Lie algebra of the isotropy subgroup $B\subset G_{\C}$ at $F$ is
$F^0\jlg$ and $F^{-1}\jlg/F^0\jlg$ is an ${\rm Ad}(B)$-invariant
subspace of $\jlg /F^0\jlg$. The corresponding $G_{\C}$-invariant
subbundle of the holomorphic tangent bundle of $\check D$ is the
\textsl{ horizontal tangent bundle\/}. A polarized VHS over a
manifold $M$ determines ---via parallel translation to a typical fiber
$V$--- a holomorphic map $\Phi\colon M \to D/\Gamma$ where $\Gamma$ is
the monodromy group (Griffiths' period map). By definition, it has
\textsl{horizontal} local liftings into $D$, \ie, its differentials
take values on the horizontal tangent bundle.
\begin{example}
Let $X$ be an $n$-dimensional, smooth projective variety, $\omega\in
H^{1,1}(X)$ a K\"ahler class. For any $k=0,\dots,2n$, the Hodge
decomposition (see~\cite{bo:griffiths-principlesAG})
\begin{equation*}
H^k(X,\C) \ = \ \bigoplus_{p+q =k} H^{p,q}(X); \quad \conj{H^{p,q}}
= H^{q,p}
\end{equation*}
determines a Hodge structure of weight $k$ by
\begin{equation*}
F^{p} \ := \ \bigoplus_{a\geq p} H^{a,k-a}.
\end{equation*}
Its restriction to the \textsl{primitive cohomology}
\begin{equation*}
H_0^{n-\ell}(X,\C) \ :=\ \{\alpha \in H^{n-\ell}(X,\C)\,:\,
\omega^{\ell+1} \cup \alpha = 0\}\,,\ \ell\geq 0,
\end{equation*}
is polarized by the form $Q_\ell(\alpha ,\beta) = Q(\alpha ,
\beta\cup \omega^\ell)$, $\alpha, \beta\in H_0^{n-\ell}(X,\C)$, and
where $Q$ denotes the signed intersection form given, for $\alpha\in
H^k(X,\C)$, $\alpha'\in H^{k'}(X,\C)$ by:
\begin{equation}\label{intersectionform}
Q(\alpha,\alpha') \ :=\ (-1)^{k(k-1)/2}\ \int_X \alpha\cup\alpha'.
\end{equation}
A family $\XX \to M$ of smooth projective varieties gives rise to a
polarized variation of Hodge structure $\VV \to M$, where $\VV_m
\cong H_0^k(\XX_m,\C)$, $m\in M$.
\end{example}
Our main concern is the asymptotic behavior of $\Phi$ near the
boundary of $M$, with respect to some compactification $\conj{M}$
where $\conj{M}-M$ is a divisor with normal crossings (the divisor at
``infinity"). Such compactifications exist, for instance, if $M$ is
quasiprojective. Near a boundary point $p\in\conj{M}-M$ we can choose
an open set $W$ such that $W\cap M \simeq (\Delta^*)^r \times
\Delta^m$ and then consider the local period map
\begin{equation}\label{localperiod}
\Phi:(\Delta^*)^r \times \Delta^m\rightarrow \DD/\Gamma.
\end{equation}
We shall also denote by $\Phi$ its lifting to the universal covering
$U^r \times \Delta^m$, where $U$ denotes the upper-half plane. We
denote by $z=(z_j)$, $t=(t_l)$ and $s=(s_j)$ the coordinates on $U^r$,
$\Delta^m$ and $(\Delta^*)^r$ respectively. By definition, we have
$s_j = e^{2\pi i z_j}$.
According to Schmid's Nilpotent Orbit Theorem \cite{ar:S-vhs}, the
singularities of $\Phi$ at the origin are, at worst, logarithmic; this
is essentially equivalent to the regularity of the connection
$\nabla$. More precisely, assuming quasi-unipotency ---this is
automatic in the geometric case--- and after passing, if necessary, to
a finite cover of $(\Delta^*)^r$, there exist commuting nilpotent
elements\footnote{Our sign convention is consistent with
\cite{ar:Del-local_behavior, ar:S-vhs} but opposite to that in
\cite{bo:CK-mirror, ar:morrison-guide}.} $N_1,\dots,N_r\in \jlg_\R$,
with $N^{k+1}=0$ and such that
\begin{equation}\label{not}
\Phi(s,t) \ =\ \exp\left(\sum_{j=1}^r\frac {\log s_j}{2\pi i}
N_j\right)\cdot \Psi(s,t)\,,
\end{equation}
where $\Psi \colon \Delta^{r+m} \to \check D$ is holomorphic.
We will refer to $N_1,\dots,N_r$ as the \textsl{ local monodromy
logarithms} and to $F_0:=\Psi(0)$ as the \textsl{limiting Hodge
filtration}. They combine to define a \textsl{nilpotent orbit}
$\{N_1,\dots,N_r; F_0\}$. We recall:
\begin{definition} With notation as above,
$\{N_1,\dots,N_r; F_0\}$ is called a nilpotent orbit if the map
\begin{equation*}
\theta(z) = \exp(\sum_{j=1}^r z_j
N_j)\cdot F_0
\end{equation*}
is horizontal and there exists $\alpha\in \R$ such that $\theta(z)
\in D$ for ${\rm Im}(z_j) > \alpha$.
\end{definition}
Theorem~\ref{th:nilporbit} below gives an algebraic characterization
of nilpotent orbits which will play a central role in the sequel.
We point out that the local monodromy is topological in nature, while
the limiting Hodge filtration depends on the choice of coordinates
$s_j$. To see this, we consider, for simplicity, the case $m=0$. A
change of coordinates compatible with the divisor structure must be,
after relabeling if necessary, of the form $(s'_1,\dots,s'_r) =
(s_1f_1(s),\dots,s_rf_r(s))$ where $f_j$ are holomorphic around $0\in
\Delta^r$ and $f_j(0) \not=0$. We then have from~(\ref{not}),
\begin{equation}\label{psicoord}
\begin{split}
\Psi'(s') &= \exp(- \frac{1}{2\pi i} \sum_{j=1}^r
\log(s'_j)N_j)\cdot \Phi(s')\\ &= \exp(- \frac{1}{2\pi i}
\sum_{j=1}^r \log(f_j)N_j) \exp(- \frac{1}{2\pi i} \sum_{j=1}^r
\log(s_j)N_j) \cdot \Phi(s)\\ &= \exp(- \frac{1}{2\pi i}
\sum_{j=1}^r \log(f_j)N_j) \cdot \Psi(s),
\end{split}
\end{equation}
and, letting $s\to 0$
\begin{equation}
\label{eq:change_F}
F'_0 = \exp (- \frac {1}{2\pi i} \sum_j \log (f_j(0)) N_j) \cdot
F_0.
\end{equation}
These constructions may also be understood in terms of Deligne's
canonical extension \cite{bo:Del-equations}. Let $\VV \to
(\Delta^*)^r \times \Delta^m$ be the local system underlying a
polarized VHS and pick a base point $(s_0,t_0)$. Given $v\in V :=
\VV_{(s_0,t_0)}$, let $v^{\flat}$ denote the multivalued flat section
of $\VV$ defined by $v$. Then
\begin{equation}\label{cansections}
\tilde v(s,t) \ :=\ \exp\left(\sum\frac {\log s_j}{2\pi i}
N_j\right)\cdot v^{\flat}(s,t)
\end{equation}
is a global section of $\VV$. The canonical extension $\conj\VV \to
\Delta^{r+m}$ is characterized by its being trivialized by sections of
the form (\ref{cansections}). The Nilpotent Orbit Theorem then
implies that the Hodge bundles $\FF^p$ extend to holomorphic
subbundles $\conj\FF^p\subset \conj\VV$. Writing the Hodge bundles in
terms of a basis of sections of the form (\ref{cansections}) yields
the holomorphic map $\Psi$. Its constant part ---corresponding to the
nilpotent orbit--- defines a polarized VHS as well.
A nilpotent linear transformation $N\in\gl(V_\R)$ defines an
increasing filtration, the \textsl{weight filtration}, $W(N) $ of $V$,
defined over $\R$ and uniquely characterized by requiring that
$N(W_l(N))\subset W_{l-2}(N)$ and that $N^l:\gr_{l}^{W(N)}\rightarrow
\gr_{-l}^{W(N)}$ be an isomorphism. It follows from
\cite[Theorem~3.3]{ar:CK-polarized} that if $N_1,\dots,N_r$ are local
monodromy logarithms arising from a polarized VHS then the weight
filtration $W(\sum \lambda_j N_j)$, $\lambda_j\in \R_{>0}$, is
independent of the choice of $\lambda_1,\dots,\lambda_r$ and,
therefore, is associated with the positive real cone $\CC \subset
\jlg_\R$ spanned by $N_1,\dots,N_r$.
A \textsl{mixed Hodge Structure} (MHS) on $V$ consists of a pair of
filtrations of $V$, $(W, F)$, $W$ defined over $\R$ and increasing,
$F$ decreasing, such that $F$ induces a Hodge structure of weight $k$
on $\gr_k^{W}$ for each $k$. Equivalently, a MHS on $V$ is a
bigrading
\begin{equation*}
V\ = \ \bigoplus I^{p,q}
\end{equation*}
satisfying $I^{p,q}\equiv \conj{I^{q,p}} \mod(\oplus_{a<p,b<q}
I^{a,b})$ (see~\cite[Theorem~2.13]{ar:CKS}). Given such a bigrading
we define: $ W_l = \oplus_{p+q \leq l} I^{p,q}$, $F^a = \oplus_{p\geq
a} I^{p,q}$. A MHS is said to \textsl{split} over $\R$ if $I^{p,q}=
\conj{I^{q,p}}$; in that case the subspaces $V_l = \oplus_{p+q = l}
I^{p,q}$ define a real grading of $W$. A map $T \in \gl(V)$ such that
$T(I^{p,q}) \subset I^{p+a,q+b}$ is called a morphism of bidegree
$(a,b)$.
A \textsl{polarized MHS} (PMHS)~\cite[(2.4)]{ar:CK-polarized} of
weight $k$ on $V_\R$ consists of a MHS $(W,F)$ on $V$, a $(-1,-1)$
morphism $N\in \lgr$, and a nondegenerate bilinear form $Q$ such that
\begin{enumerate}
\item $N^{k+1}=0$,
\item $W = W(N)[-k]$, where $W[-k]_j = W_{j-k}$,
\item $Q(F^a,F^{k-a+1}) = 0$ and,
\item the Hodge structure of weight $k+l$ induced by $F$ on
$\ker(N^{l+1}:\gr_{k+l}^{W}\rightarrow \gr_{k-l-2}^{W})$ is
polarized by $Q(\cdot,N^l \cdot)$.
\end{enumerate}
\begin{theorem} \label{th:nilporbit}
Given a nilpotent orbit $\theta(z) = \exp(\sum_{j=1}^r z_j N_j)\cdot
F $, the pair $(W(\CC),F)$ defines a MHS polarized by every $N\in
\CC$. Conversely, given commuting nilpotent elements
$\{N_1,\ldots,N_r\}\in \lgr$ with the property that the weight
filtration $W(\sum \lambda_j N_j)$, $\lambda_j\in \R_{>0}$, is
independent of the choice of $\lambda_1,\dots,\lambda_r$, \ if $F
\in \DC$ is such that $(W(\CC),F)$ is polarized by every element
$N\in \CC$, then the map $\theta(z) = \exp(\sum_{j=1}^r z_j
N_j)\cdot F$ is a nilpotent orbit. Moreover, if $(W(\CC),F)$ splits
over $\R$, then $\theta(z) \in D$ for ${\rm Im}(z_j) > 0$.
\end{theorem}
The first part of Theorem~\ref{th:nilporbit} was proved by Schmid
\cite[Theorem~6.16]{ar:S-vhs} as a consequence of his $SL_2$-orbit
theorem. The converse is Proposition~4.66 in \cite{ar:CKS}. The
final assertion is a consequence of
\cite[Proposition~2.18]{ar:CK-polarized}.
\begin{example}\label{ex:totalcohomology}
Let $X$ be an $n$-dimensional, smooth projective variety. Let $V =
H^*(X,\C)$, $V_\R = H^*(X,\R)$. The bigrading $I^{p,q} :=
H^{n-p,n-q}(X)$ defines a MHS on $V$ which splits over $\R$. The
weight and Hodge filtrations are then
\begin{equation*}
W_l \ =\ \bigoplus_{d\geq 2n-l} H^d(X,\C),\quad F^p \ =\
\bigoplus_{s}\bigoplus_{r\leq n-p}H^{r,s}(X).
\end{equation*}
Given a K\"ahler class $\omega\in H^{1,1}(X,\R):= H^{1,1}(X) \cap
H^2(X,\R)$, let $L_{\omega}\in \gl(V_\R)$ denote multiplication by
$\omega$. Note that $L_{\omega}$ is an infinitesimal automorphism
of the form (\ref{intersectionform}) and is a $(-1,-1)$ morphism of
$(W,F)$. Moreover, the Hard Lefschetz Theorem and the Riemann
bilinear relations are equivalent to the assertion that $L_\omega$
polarizes $(W,F)$. Let $\KK \subset H^{1,1}(X,\R)$ denote the
K\"ahler cone and
\begin{equation*}
\KK_\C := H^{1,1}(X,\R) \oplus i \KK \subset H^2(X,\C)
\end{equation*}
the complexified K\"ahler cone. It then follows from
Theorem~\ref{th:nilporbit} that for every $\xi\in \KK_\C$, the
filtration $\exp(L_\xi)\cdot F$ is a Hodge structure of weight $n$
on $V$ polarized by $Q$. The map $\, \xi\in \KK_\C \mapsto
\exp(L_\xi)\cdot F \,$ is the period map (in fact, the nilpotent
orbit) of a variation of Hodge structure over $\KK_\C$. Note that
we can restrict the above construction to $V = \oplus_p H^{p,p}(X)$;
this is the case of interest in mirror symmetry.
\end{example}
\begin{remark}
The notion of nilpotent orbit is closely related to that of
Lefschetz modules introduced by Looijenga and
Lunts~\cite{ar:LL-lefschetz_modules}. Indeed, it follows from
\cite[Proposition~1.6]{ar:LL-lefschetz_modules} that if $(W,F)$ is a
MHS, polarized by every $N$ in a cone $\CC$, and $\scA$ denotes the
linear span of $\CC$ in $\lgr$, then $V$ is a Lefschetz module of
$\scA$.
\end{remark}
Let now $\Phi$ be as in (\ref{localperiod}), $\{N_1,\dots,N_r;F_0\}$
the associated nilpotent orbit and $(W(\CC),F_0)$ the \textsl{limiting
mixed Hodge structure}. The bigrading $I^{*,*}$ of $V$ defined by
$(W(\CC),F_0)$ defines a bigrading $I^{*,*}\jlg$ of the Lie algebra
$\jlg$ associated with the MHS $(W(\CC)\jlg,F_0\jlg)$. Set
\begin{equation}
\label{eq:def_pa}
\gp_a \ := \ \bigoplus_{q}I^{a,q}\jlg \quad \text{ and }\quad \jlg_-
\ := \ \bigoplus_{a\leq
-1}\gp_a.
\end{equation}
The nilpotent subalgebra $\jlg_-$ is a complement of the stabilizer
subalgebra at $F_0$. Hence $(\jlg_-, X \mapsto \exp(X)\cdot F_0)$
provides a local model for the $\GC$-homogeneous space $\DC$ near
$F_0$ and we can rewrite (\ref{not}) as:
\begin{equation}\label{gamma}
\Phi(s,t) \ =\ \exp\left(\sum_{j=1}^r\frac {\log s_j}{2\pi i}
N_j\right)\ \exp \Gamma(s,t)\cdot F_0\,,
\end{equation}
where $\Gamma\colon \Delta^{r+m} \to \jlg_-$ is holomorphic and
$\Gamma(0) = 0$. The lifting of $\Phi$ to $U^r\times \Delta^m$ may
then be expressed as:
\begin{equation*}
\Phi(z,t) = \exp X(z,t)\cdot F_0
\end{equation*}
with $X:U^r\times \Delta^m\rightarrow \jlg_-$ holomorphic. Setting
$E(z,t) := \exp X(z,t)$, the horizontality of $\Phi$ is then expressed
by:
\begin{equation}
\label{eq:horiz}
E^{-1}\, dE = dX_{-1} \in \gp_{-1}\otimes T^*(U^r\times \Delta^m).
\end{equation}
The following explicit description of period mappings near infinity is
proved in ~\cite[Theorem~2.8]{ar:CK-luminy}.
\begin{theorem}\label{th:2.8}
Let $\{N_1,\ldots,N_r;F_0\}$ be a nilpotent orbit and
$\Gamma:\Delta^r\times \Delta^m \rightarrow \jlg_-$ be holomorphic,
such that $\Gamma(0,0) = 0$. If the map
\begin{equation}
\label{eq:2.8_form}
\Phi(z,t)=\exp(\sum z_j N_j)\ \exp(\Gamma(s,t))\cdot F_0
\end{equation}
is horizontal (\ie, (\ref{eq:horiz}) is satisfied), then $\Phi(z,t)$
is a period mapping.
\end{theorem}
Given a period mapping as in (\ref{eq:2.8_form}), let $\Gamma_{-1}(s,t)$
denote the $\gp_{-1}$-component of $\Gamma$. Then,
\begin{equation*}
X_{-1}(z,t)\ =\ \sum_{j=1}^r z_j N_j+\Gamma_{-1}(s,t)\,, \text{ with }
s_j=e^{2\pi i z_j},
\end{equation*}
and it follows from (\ref{eq:horiz}) that
\begin{equation} \label{eq:integcond}
dX_{-1}\wedge dX_{-1}=0
\end{equation}
The following theorem shows that this equation characterizes period
mappings with a given nilpotent orbit.
\begin{theorem}
\label{th:improved_2.8}
Let $R:\Delta^r \times \Delta^m \rightarrow \gp_{-1}$ be a
holomorphic map with $R(0)=0$. Let $X_{-1}(z,t) = \sum_{j=1}^r z_j
N_j+R(s,t)$, $s_j = e^{2\pi i z_j}$, and suppose that the
differential equation (\ref{eq:integcond}) holds. Then, there
exists a unique period mapping (\ref{gamma}) defined in a
neighborhood of the origin in $\Delta^{r+m}$ and such that
$\Gamma_{-1} = R$.
\end{theorem}
\begin{proof}
To prove uniqueness, we begin by observing that if $\Phi$ and
$\Phi'$ are period mappings with the same associated nilpotent orbit
and $\Gamma_{-1}(s,t) = \Gamma_{-1}'(s,t)$ then, for any $v \in
F^p$, we may consider the sections $\nu(s,t) = E(s,t)\cdot
v^\flat(s,t)$ and $\nu'(s,t) = E'(s,t)\cdot v^\flat(s,t)$ of the
canonical extension $\conj{\mathcal V}$. Clearly, $\nu(s,t)\in
{\mathcal F}_{(s,t)}^p$ and $\nu'(s,t)\in {\mathcal F'}_{(s,t)}^p$.
On the other hand, since $\Gamma_{-1}(s,t) = \Gamma_{-1}'(s,t)$, it
follows that $E_{-1}(s,t) = E_{-1}'(s,t)$ and, consequently,
$\nu(s,t)-\nu'(s,t)$ is a $\nabla$-flat section which extends to the
origin and takes the value zero there. Hence, $\nu(s,t)-\nu'(s,t)$
is identically zero and ${\mathcal F}_{(s,t)}^p = ({\mathcal
F}_{(s,t)}')^p$ for all values of $(s,t)$.
To complete the proof of the Theorem it remains to show the
existence of a period mapping with given nilpotent orbit and
$\Gamma_{-1}(s,t) = R(s,t)$. This amounts to finding a solution to
the differential equation (\ref{eq:horiz}) with
\begin{equation*}
X_{-1}(z,t)\ =\ \sum_{j=1}^r z_j N_j+ R(s,t)\,, \text{ with }
s_j=e^{2\pi i z_j},
\end{equation*}
assuming that the integrability condition (\ref{eq:integcond}) is
satisfied. Set $G(s,t) = \exp \Gamma(s,t)$ and $\Theta= d
(\sum_{j=1}^r z_j N_j)$. Then (\ref{eq:horiz}) may be rewritten as
\begin{equation}
\label{eq:e5}
d G = [G,\Theta] + G d G_{-1} \text{ with } G(0,0)=\idM,
\end{equation}
where $\idM$ denotes the identity, while the
condition~(\ref{eq:integcond}) takes the form:
\begin{equation}
\label{eq:e6}
d G_{-1} \wedge \Theta + \Theta \wedge d G_{-1} + d G_{-1}\wedge d
G_{-1} = 0.
\end{equation}
By considering the $\gp_{-l}$-graded components of~(\ref{eq:e5}) we
obtain a sequence of equations:
\begin{equation}
\label{eq:e7}
d G_{-l} = [G_{-l+1}, \Theta] + G_{-l+1} d G_{-1}, \quad
G_{-l}(0,0)=0, \quad l\geq 2.
\end{equation}
Assume inductively that, for $l\geq 2$, we have constructed
$G_{-l+1}$ satisfying~(\ref{eq:e7}) and such that
\begin{equation*}
d G_{-l+1} \wedge \Theta + \Theta \wedge d G_{-l+1} + d
G_{-l+1}\wedge d G_{-1} = 0.
\end{equation*}
Then, the initial value problem
\begin{equation*}
d G_{-l} = [G_{-l+1}, \Theta] + G_{-l+1} d G_{-1},\quad
G_{-l}(0,0) = 0,
\end{equation*}
has a solution which verifies
\begin{eqnarray*}
\lefteqn{d G_{-l} \wedge \Theta + \Theta \wedge d G_{-l} + d
G_{-l} \wedge d G_{-1} =}\\ &=& [ G_{-l+1}, \Theta] \wedge
\Theta + G_{-l+1} d G_{-1} \wedge \Theta + \Theta \wedge
[G_{-l+1}, \Theta] + \Theta \wedge G_{-l+1} d G_{-1} + \\ & &
\mbox{} + [G_{-l+1}, \Theta] \wedge d G_{-1} + G_{-l+1} d G_{-1}
\wedge d G_{-1} \\ &=& -\Theta \wedge G_{-l+1} \Theta +G_{-l+1} d
G_{-1} \wedge \Theta + \Theta \wedge G_{-l+1} \Theta + \Theta
\wedge G_{-l+1} d G_{-1} +\\ & & \mbox{} + G_{-l+1} \Theta \wedge
d G_{-1} + G_{-l+1} d G_{-1} \wedge d G_{-1} - \Theta \wedge
G_{-l+1} d G_{-1} \\ &=& G_{-l+1} (d G_{-1} \wedge \Theta + \Theta
\wedge d G_{-1} + d G_{-1} \wedge d G_{-1}) = 0.
\end{eqnarray*}
Thus we may, inductively, construct a solution of
~(\ref{eq:e6}). Theorem~\ref{th:2.8} now implies that the map
\begin{equation*}
\Phi(z,t) = \exp(\sum_{j=1}^r z_j N_j)\ G(s,t)\cdot F_0
\end{equation*}
is the desired period map.
\end{proof}
\begin{remark}
The uniqueness part of the argument is contained in Lemmas~2.8
and~2.9 of \cite{ar:CDK}, while the existence proof is contained in
the unpublished manuscript \cite{prep:C-addenda}. A particular case
of Theorem~\ref{th:improved_2.8} is given in
\cite[Theorem~11]{ar:Del-local_behavior}; a generalization to the
case of variations of MHS appears in \cite{ar:greg-higgs}.
\end{remark}
\section{Canonical Coordinates}
\label{sec:canonical_coordinates}
The asymptotic data of a polarized variation of Hodge structure over
an open set $W \simeq (\Delta^*)^r \times \Delta^m$ depends on the
choice of coordinates on the base. We have already observed that the
local monodromy logarithms $N_j$ are independent of coordinates and
have shown in (\ref{eq:change_F}) how the limiting Hodge filtration
changes under a coordinate transformation. Here we will discuss the
dependence of the holomorphic function $\Gamma \colon (\Delta^*)^r
\times \Delta^m \to \gp_{-1}$ and show that, in special cases, there
is a natural choice of coordinates. This choice will be seen to agree
with that appearing in the mirror symmetry setup and which has already
been given a Hodge-theoretic interpretation by Deligne
\cite{ar:Del-local_behavior}. These canonical coordinates may also be
interpreted, in the case of families of Calabi-Yau threefolds as the
coordinates where the Picard-Fuchs equations take on a certain
particularly simple form (\cite[Prop. 5.6.1]{bo:CK-mirror}). To
simplify the discussion we will restrict our discussion to the case
$m=0$. The general case, which follows easily, will not be needed in
the sequel.
Since we are required to preserve the divisor structure at the boundary,
we want to study the behavior of the asymptotic data under coordinate
changes of the form
\begin{equation}
\label{eq:coordinate_transformation}
s'_j = s_j f_j(s)
\end{equation}
where the functions $f_j$ are holomorphic in a neighborhood of $0
\in \Delta^r$ and $f_j(0)\neq 0$.
Given a PVHS over $(\Delta^*)^r$ and a choice of local coordinates
$(s_1,\dots,s_r)$ around
$0$, we write the associated period map as in (\ref{gamma}):
\begin{equation*}
\Phi(s) \ =\
\exp(\sum_{j=1}^r \frac{\log s_j}{2\pi i} N_j)
\exp(\Gamma(s)) \cdot F_0
\end{equation*}
Given another system of coordinates $s' = (s'_1,\dots,s'_r)$ as in
(\ref{eq:coordinate_transformation}), let $F'_0$ and $\Gamma'$ denote
the corresponding asymptotic data. By (\ref{eq:change_F}), $F'_0 =
{\mathcal M}\cdot F_0$, where
\begin{equation*}
{\mathcal M}\ := \ \exp(-\frac{1}{2\pi
i}\sum_{j=1}^r\log f_j(0)\,N_j).
\end{equation*}
\begin{prop}\label{prop:change_coordinates}
Under a coordinate change as in
(\ref{eq:coordinate_transformation}):
\begin{equation}
\label{eq:change_gamma}
{\mathcal M}^{-1}\exp(\Gamma'(s')) {\mathcal M} \ =\
\exp\left(-\frac{1}{2\pi i} \sum_{j=1}^r \log
\frac{f_j(s)}{f_j(0)}\,N_j\right) \exp(\Gamma(s)).
\end{equation}
\end{prop}
\begin{proof} Let $W$ denote the filtration $W(N)[-k]$, where
$N$ is an arbitrary element in the cone $\CC$ positively spanned by
$N_1,\dots,N_r$. Note that $\MM$ leaves $W$ invariant. Moreover,
since the monodromy logarithms are $(-1,-1)$-morphisms of the mixed
Hodge structure $(W,F_0)$ it follows easily that
\begin{equation*}
I^{a,b}(W,F'_0) \ =\ \MM\cdot I^{a,b}(W,F_0),
\end{equation*}
where $I^{*,*}(W,F_0)$ denotes the canonical bigrading of the MHS.
This implies that the associated bigrading of the Lie algebra $\jlg$
and, in particular, that the subalgebra $\jlg_-$ defined in
(\ref{eq:def_pa}) are independent of the choice of coordinates.
According to (\ref{psicoord}), $\Psi'(s') \ =\ \exp\left(\frac
1{2\pi i}\sum_j {\log f_j(s)} N_j\right)\cdot \Psi(s)$, therefore
\begin{equation}\label{intermediate}
\exp \Gamma'(s')\, \MM\cdot F_0 \ =\ \exp\left(\frac {1}{2\pi
i}\sum_{j=1}^r \log f_j(s)\, N_j\right)\ \exp \Gamma(s)\cdot F_0.
\end{equation}
This identity, in turn, implies (\ref{eq:change_gamma}) since the
group elements in both sides of (\ref{intermediate}) lie in
$\exp(\jlg_-)$.
\end{proof}
\begin{corollary}\label{cor:change_coordinates-1}
With the same notation of Proposition~\ref{prop:change_coordinates},
\begin{equation}
\label{eq:change_w1}
{\mathcal M}^{-1}\Gamma'_{-1}{\mathcal M}\ = \ -\frac{1}{2\pi i}
\sum_j \log \frac{f_j(s)}{f_j(0)}\, N_j + \Gamma_{-1}.
\end{equation}
\end{corollary}
\begin{proof}
This follows considering the $\gp_{-1}$-component
in~(\ref{eq:change_gamma}), given the observation that this subspace
is invariant under coordinate changes.
\end{proof}
Up to rescaling, we may assume that our coordinate change
(\ref{eq:coordinate_transformation}) satisfies $f_j(0) =1$,
$j=1\dots,r$. Such changes will be called {\sl simple}. In this case
$\MM = \idM$, $F'_0 = F_0$, and the transformation
(\ref{eq:change_w1}) is just a translation in the direction of the
nilpotent elements $N_j$. Thus, whenever the subspace spanned by
$N_1,\dots,N_r$ has a natural complement in $\gp_{-1}$ we will be able
to choose coordinates, unique up to scaling, such that $\Gamma_{-1}$
takes values in that complement. This is the situation in the
variations of Hodge structure studied in mirror symmetry. In this
context, one analyzes the behavior of PVHS near some special boundary
points. They come under the name of ``large radius limit points''
(see~\cite[\S 6.2.1]{bo:CK-mirror}) or ``maximally unipotent boundary
points'' (see~\cite[\S 5.2]{bo:CK-mirror}). For our purposes, we have
\begin{definition}\label{maxunip}
Given a PVHS of weight $k$ over $(\Delta^*)^r$ whose monodromy is
unipotent, we say that $0\in \Delta^r$ is a \textsl{maximally
unipotent boundary point} if
\begin{enumerate}
\item $\dim I^{k,k} = 1$, $\dim I^{k-1,k-1} = r$ and $\dim I^{k,k-1}
= \dim I^{k-2,k}= 0$, where $I^{a,b}$ is the bigrading associated
to the limiting MHS and,
\item $\vspan_\C(N_1(I^{k,k}),\ldots,N_r(I^{k,k})) = I^{k-1,k-1}$,
where $N_j$ are the monodromy logarithms of the variation.
\end{enumerate}
\end{definition}
Under these conditions, we may identify $\vspan_\C(N_1,\dots,N_r)
\cong \homom(I^{1,1},I^{0,0})$. Hence, denoting by $\rho\colon
\gp_{-1} \to \homom(I^{1,1},I^{0,0})$ the restriction map, the
subspace $K = \ker(\rho)$ is a canonical complement of
$\vspan_\C(N_1,\ldots,N_r)$ in $\gp_{-1}$.
Note that both, the notion of maximally unipotent boundary point and
the complement $K$ are independent of the choice of basis.
\begin{definition}\label{def:cancoord}
Let $\VV \to \poly$ be a PVHS having the origin as a maximally
unipotent boundary point. A system of local coordinates
$(q_1,\dots,q_r)$ is called {\sl canonical} if the associated
holomorphic function $\Gamma_{-1}$ takes values in $K$.
\end{definition}
\begin{prop}\label{prop:special_coordinates}
Let $\VV \to \poly$ be a PVHS having the origin as a maximally
unipotent boundary point. Then there exists, up to scaling, a
unique system of canonical coordinates.
\end{prop}
\begin{proof}
Let $s=(s_1,\dots,s_r)$ be an arbitrary system of coordinates around
$0$. We can write
\begin{equation*}
\rho(\Gamma(s)) \ =\ \sum_{j=1}^r \gamma_j(s)\,N_j,
\end{equation*}
where $\gamma_j(s)$ are holomorphic in a neighborhood of $0\in
\Delta^r$ and $\gamma_j(0)=0$. The transformation formula
(\ref{eq:change_w1}) now implies that the coordinate system
\begin{equation}\label{extension}
q_j \ :=\ s_j \exp(2\pi i \gamma_j(s))
\end{equation}
is canonical. Moreover, that same formula shows that it is unique
up to scaling.
\end{proof}
In \cite{ar:Del-local_behavior}, Deligne observed that a variation of
Hodge structure whose limiting MHS is of Hodge-Tate type defines,
together with the monodromy weight filtration, a variation of mixed
Hodge structure. In this context, the holomorphic functions
$q_{j}(s)$ defined by (\ref{extension}) constitute part of the
extension data of this family of mixed Hodge structures. He shows,
moreover, that they agree with the special coordinates studied in
\cite{ar:CDGP-pair}, \cite{ar:morrison-guide}, and
\cite{ar:morr-picard-fuchs} for families of Calabi-Yau manifolds in
the vicinity of a maximally unipotent boundary point. We sketch this
argument for the sake of completeness.
Given a coordinate system $s=(s_1,\dots,s_r)$, let $(W,F_0)$ be the
limiting MHS of weight $k$ and choose $e^0 \in I^{0,0}$. Let $e^k \in
I^{k,k}$ be such that $Q(e^0,e^k) = (-1)^k$. Since the origin is a
maximally unipotent boundary point, there exists a basis $e_1^{
1},\dots,e_r^{ 1}$ of $I^{1 , 1}$ such that $N_j(e_l^{ 1}) =
\delta_{jl} e^0$. We can define a (multi-valued) holomorphic section
of $\FF^k$ by
\begin{equation*}
\omega(s)\ :=\ \exp(\sum_{j=1}^r \frac{\log s_j}{2\pi i} N_j)
\exp(\Gamma(s))\cdot e^k\,.
\end{equation*}
In the geometric setting of a family of varieties $X_s$ the
coefficients
\begin{equation*}
h_0(s) \ :=\ -Q(e^0, \omega(s)) \quad\hbox{and}\quad h_j(s) \ :=\
Q(e^1_j, \omega(s))
\end{equation*}
may be interpreted as integrals $\int_\alpha \omega(s)$ over
appropriate cycles $\alpha \in H_k(X_{s_0})$ on the typical fiber.
Clearly, our assumptions imply that $h_0(s) = (-1)^{k+1}$ and
\begin{equation*}
\begin{split}
h_j(s)\ &=\ Q(e^1_j, (\sum_l \frac {\log s_l}{2\pi i} N_l +
\Gamma_{-1}(s))\cdot e^k) \\ &= \ - \frac {\log s_j}{2\pi i}
Q(e^0,e^k) - Q(\Gamma_{-1}(s)\cdot e^1_j,e^k)\\ &=\ - \frac {\log
s_j}{2\pi i} Q(e^0,e^k) - Q(\sum_l\gamma_l(s) N_l e^1_j,e^k)\\ &=\
(-1)^{k+1} (\frac {\log s_j}{2\pi i} + \gamma_j(s)).
\end{split}
\end{equation*}
Therefore, $q_j(s) = \exp(2\pi i h_j(s)/h_0(t))$, which agrees with
Morrison's geometric description of the canonical coordinates in
\cite[\S 2]{ar:morr-picard-fuchs}.
\section{Graded Frobenius Algebras and Potentials}
\label{sec:frobenius}
In this section we will abstract the basic properties of the
cup product in the cohomology subalgebra $\oplus_p H^{p,p}(X)$
for a smooth projective variety $X$ and show that this product
structure may be encoded in a single homogeneous polynomial of
degree $3$.
We recall that $(V,*,e_0,\CB)$ is called a Frobenius algebra if
$(V,*)$ is an associative, commutative $\C$-algebra with unit $e_0$,
and $\CB$ is a nondegenerate symmetric bilinear form such that
$\CB(v_1 * v_2, v_3) = \CB(v_1, v_2 * v_3)$. The algebra is said to
be real if $V$ has a real structure $V_\R$, $e_0\in V_\R$ and both $*$
and $\CB$ are defined over $\R$. Throughout this paper we will be
interested in {\sl graded}, real Frobenius algebras of weight $k$. By
this we mean that $V$ has an even grading
\begin{equation*}
V\ =\ \bigoplus_{p=0}^k V_{2p},
\end{equation*}
defined over $\R$, and such that:
\begin{enumerate}
\item $V_0 \ \cong \ \C$,
\item $(V,*)$ is a graded algebra,
\item $\CB(V_{2p}, V_{2q}) \ =\ 0$ if $p+q \not= k$.
\end{enumerate}
The product structure on a Frobenius algebra $(V,*,e_0,\CB)$ may be
encoded in the trilinear function $\ti{\phi}_0: V\times V \times V
\rightarrow \C$
\begin{equation*}
\ti{\phi}_0(v_1,v_2,v_3)\ :=\ \CB(e_0,v_1*v_2*v_3),
\end{equation*}
or, after choosing a graded basis $\{e_0,\ldots,e_m\}$ of $V$, in the
associated cubic form:
\begin{equation}\label{eq:phi_0}
\phi_0(z_0,\ldots,z_m)\ :=\ \frac{1}{6}\ti{\phi}_0(\gamma, \gamma,
\gamma)\ =\ \frac{1}{6} \CB(e_0, \gamma^3),\quad \gamma \ =\
\sum_{a=0}^m z_a e_a.
\end{equation}
Let $\{e_a^\CB\ :=\ \sum_b h_{ba}e_b\}$ denote the $\CB$-dual basis of
$\{e_a\}$.
\begin{theorem}\label{thm:prod_pot}
The cubic form $\phi_0$ defined by~(\ref{eq:phi_0}) is (weighted)
homogeneous of degree $k$ with respect to the grading defined by
$\deg z_a = \deg e_a$, and satisfies the algebraic relation:
\begin{equation}
\label{eq:prod_pot_2}
\sum_{d,f} \frac{\del^3 \phi_0}{\del z_a \del z_b \del z_d} h_{df}
\frac{\del^3 \phi_0}{\del z_f \del z_c \del z_g} \ = \ \sum_{d,f}
\frac{\del^3 \phi_0}{\del z_b \del z_c \del z_d} h_{df}
\frac{\del^3 \phi_0}{\del z_a \del z_f \del z_g}.
\end{equation}
The product $*$ and the bilinear form $\CB$ may be expressed in
terms of $\phi_0$ by
\begin{eqnarray}
\label{eq:prod_pot_1}
\CB(e_a, e_b) &=& \frac{\del^3 \phi_0}{\del z_0 \del z_a \del z_b}
\\
\label{eq:prod_pot_3}
e_a * e_b &=& \sum_c \frac{\del^3 \phi_0}{\del z_a \del z_b \del
z_c} e_c^\CB\,.
\end{eqnarray}
Conversely, let $V$ be an evenly graded vector space, $V_0\cong \C$,
$\{e_0,\ldots, e_m\}$ a graded basis of $V$, and $\phi_0 \in
\C[z_0,\dots,z_m]$ a cubic form, homogeneous of degree $k$ relative
to the grading $\deg z_a = \deg e_a$, satisfying
(\ref{eq:prod_pot_2}). Then, if the bilinear symmetric form defined
by (\ref{eq:prod_pot_1}) is non-degenerate, the product
(\ref{eq:prod_pot_3}) turns $(V,*,e_0,\CB)$ into a graded Frobenius
algebra of weight $k$.
\end{theorem}
\begin{proof}
We note, first of all, that the quasi-homogeneity of $\phi_0$
follows from the assumption that $*$ is a graded product. Moreover,
(\ref{eq:prod_pot_1}) is a consequence of the fact that $e_0$ is the
unit for $*$, while~(\ref{eq:prod_pot_2}) is the associativity
condition for $*$. On the other hand, (\ref{eq:phi_0}) and the fact
that $(V,*,e_0,\CB)$ is Frobenius imply that
\begin{equation*}
\frac{\del^3 \phi_0}{\del z_a \del z_b \del z_c} \ =\
\CB(e_0,e_a*e_b*e_c) \ =\ \CB(e_a*e_b,e_c)
\end{equation*}
and (\ref{eq:prod_pot_3}) follows.
The converse is immediate since~(\ref{eq:prod_pot_3}) defines a
commutative structure whose associativity follows
from~(\ref{eq:prod_pot_2}); the quasi-homogeneity assumption implies
that the product is graded; $e_0$ is the unit because
of~(\ref{eq:prod_pot_1}). The compatibility between $\CB$ and $*$
comes from $\CB(e_a*e_b, e_c) = \frac{\del^3 \phi_0}{\del z_a \del
z_b \del z_c} = \CB(e_a, e_b * e_c)$.
\end{proof}
\begin{example}\label{ex:classical_potential_4}
Let $(V,*,e_0,\CB)$ be a graded real Frobenius algebra of weight
four. Setting $\dim V_{2} = r$ and $\dim V_{4} = s$, we have $\dim
V = 2r + s + 2 :=m + 1$. We choose a basis $\{T_0,\dots,T_m\}$ of
$V$ as follows: $T_0 = e_0 \in V_0$ is the multiplicative unit,
$\{T_1,\dots,T_r\}$ is a real basis of $V_2$, $\{T_{r+1},\dots,
T_{r+s}\}$ is a basis of $V_4$ such that $B(T_{r+a},T_{r+b}) =
\delta_{a,b}$. Finally, $\{T_{r+s+1},\dots, T_{m-1}\}$ and $T_m$
are chosen as the $\CB$-duals of $\{T_1,\dots,T_r\}$ and $T_0$ in
$V_6$ and $V_8$, respectively. We will say that such a basis is
{\sl adapted} to the graded Frobenius structure.
It now follows from (\ref{eq:phi_0}), that with respect to such a
basis, the polynomial $\phi_0$ is given by
\begin{equation}
\label{eq:classical_potential_4}
\phi_0(z) = \frac{1}{2} z_0^2 z_m + z_0 \sum_{j=1}^r z_{j}
z_{r+s+j}+ \frac{1}{2} z_0 \sum_{a=1}^s z_{r+a}^2 +
\frac{1}{2} \sum_{a=1}^s z_{r+a} P^a(z_{1},\ldots,z_{r}),
\end{equation}
where $P^a(z_{1},\ldots,z_{r}) = \sum_{j,k=1}^r P^a_{jk} z_jz_k$ are
homogeneous polynomials of degree $2$ determined by:
\begin{equation}\label{polynomials}
P^a_{jk} = \CB(T_{r+a}, T_{j} * T_{k}).
\end{equation}
Specializing to the case when $V = \oplus_{p=0}^4 H^{p,p}(X)$ for a
smooth, projective fourfold $X$, endowed with the cup product and
the intersection form
\begin{equation*}
\CB(\alpha,\beta) \ :=\ \int_X \alpha \cup \beta\,,
\end{equation*}
we obtain
\begin{equation*}
\ti{\phi}_0^{cup}(\omega_1,\omega_2,\omega_3) = \int_X \omega_1
\cup \omega_2 \cup \omega_3.
\end{equation*}
\end{example}
In order to complete the analogy with the structure deduced from the
cup product on the cohomology of a smooth projective variety we need
to require that the Lefschetz Theorems be satisfied. Given $w \in
V_2$, let $L_w \colon V \to V$, denote the multiplication operator
$L_w(v) = w * v$. The fact that $*$ is associative and commutative
implies that the operators $L_w$, $w\in V_2$, commute, while the
assumption that $*$ is graded implies that $L_w$ is nilpotent and
$L_w^{k+1} = 0$. Given a basis $w_1,\dots,w_r$ of $V_2$, we can view
$V$ as a module over the polynomial ring $\jgU :=
\C[L_{w_1},\dots,L_{w_r}]$. Clearly, if $V = \jgU\cdot e_0$, then the
product structure $*$ may be deduced from the action of $\jgU$. This
conclusion is also true for $k=3$ or $k=4$ without further conditions,
as can be checked explicitly.
If $V$ underlies a real, graded Frobenius algebra of weight $k$, we
can define on $V$ a mixed Hodge structure of Hodge-Tate type by
$I^{p,q} = 0$ if $p\not=q$, and $I^{p,p} = V_{2(k-p)}$. If we set
\begin{equation}
\label{eq:Q_from_B}
Q(v_a, v_b) = (-1)^a \CB(v_a,v_b) \text{ if } v_a \in V_{2a},
\end{equation}
then $Q$ has parity $(-1)^k$ and for $w\in V_2 \cap V_\R$, the
operator $L_w$ is an infinitesimal automorphism of $Q$ and a $(-1,-1)$
morphism of the MHS. We also observe, that in the geometric case
---such as in Example~\ref{ex:classical_potential_4}--- this
construction agrees with that in Example~\ref{ex:totalcohomology}.
\begin{definition}\label{polarizationforfrobenius}
An element $w\in V_{2}\cap V_\R$ is said to \textsl{polarize}
$(V,*,e_0,\CB)$ if $(I^{*,*},Q,L_w)$ is a polarized MHS. A real,
graded Frobenius algebra is said to be polarized if it contains a
polarizing element. In this case, the cubic form $\phi_0$ is called
a {\sl classical potential}.
\end{definition}
By Theorem~\ref{th:nilporbit}, a polarizing element $w$ determines a
one dimensional nilpotent orbit $(W(L_w),F)$, where $W_l(L_w) =
\oplus_{2j\geq 2k-l} V_{2j}$ is the weight filtration of $L_w$ and
$F^p = \oplus_{j\leq k-p} V_{2j}$.
Given a polarizing element $w$, the set of polarizing elements is an
open cone in $V_{2} \cap V_\R$. Then, it is possible to choose a basis
$w_1, \ldots, w_r$ of $V_{2}\cap V_\R$ spanning a simplicial cone
${\mathcal C}$ contained in the closure of the polarizing cone and
with $w\in {\mathcal C}$. Such a choice of a basis of $V_2$ will be
called a {\sl framing} of the polarized Frobenius algebra.
Since the weight filtration is constant over all the elements $L_w$
for $w\in {\mathcal C}$, it follows from Theorem~\ref{th:nilporbit}
that $(W({\mathcal C}),F^*)$ is a nilpotent orbit. Hence, we can
define a polarized VHS on $(\Delta^*)^r$ whose period mapping is given
by
\begin{equation}\label{theta}
\theta(q_1,\dots,q_r) \ =\ \exp(\sum_{j=1}^r z_j L_{w_j}) \cdot F\
;\quad q_j = e^{2\pi i z_j}\,.
\end{equation}
Note that the origin is a maximally unipotent boundary point in the
sense of Definition~\ref{maxunip}.
We conclude this section showing that in the weight-four case,
maximally unipotent, Hodge-Tate, nilpotent orbits yield graded
Frobenius algebras. In the weight-three case this is done in
\cite[Example~14]{ar:Del-local_behavior}.
\begin{prop}\label{prop:constant_prod_from_nilpotent}
Let $({N}_1,\ldots, {N}_r;F)$ be a weight-four nilpotent orbit,
polarized by $Q$, whose limiting MHS is Hodge-Tate. Suppose that $
\dim I^{4,4} =1$ and choose a non-zero element ${e_0} \in I^{4,4}
\cap V_\R$; let $e^*_0 \in I^{0,0} \cap V_\R$ be such that
$Q(e_0,e_0^*)=1$. Assume, moreover, that
$\{N_1(e_0),\dots,N_r(e_0)\}$ are a basis of $ I^{3,3}$. Let $\CB$
be obtained from $Q$ as in~(\ref{eq:Q_from_B}). Then, there exists
a unique product $*$ on $V$ with unit $e_0$ such that
\begin{eqnarray}\label{eq:prod_from_NO_1}
N_j(e_0) * v &:=& N_{j}(v)\,, \text{ for } v\in V,\, j=1,\dots,r
\\
\label{eq:prod_from_NO_2}
v_1 * v_2 &:=& \CB(v_1,v_2)\, e_0^*\,, \text{ for } v_1,v_2\in
I^{2,2}
\end{eqnarray}
Furthermore, $(V,*,e_0,\CB)$ is a graded, polarized, real Frobenius
algebra.
\end{prop}
\begin{proof}
It is clear that~(\ref{eq:prod_from_NO_1})
and~(\ref{eq:prod_from_NO_2}) define a graded product whose unit is
$e_0$. Commutativity follows immediately from the symmetry of $Q$
and the commutativity of the operators ${ N}_j$.
There are two non-trivial cases to check in order to prove the
associativity of the product. When all three factors lie in
$I^{3,3}$ this follows, again, from the commutativity of the
operators $N_j$. On the other hand, given $v\in I^{2,2}$:
\begin{equation*}
\begin{split}
(N_j(e_0)*N_k(e_0))*v \ &=\ \CB(N_j(e_0)*N_k(e_0),v) \, e_0^*\ =\
\CB({N}_j({N}_k(e_0)) , v)\, e_0^*\\ &=\ \CB({N}_k({N}_j(e_0)) ,
v) \, e_0^*\ =\ \CB(e_0, {N}_j({N}_k(v))) \, e_0^*\\ &=\ \CB(e_0,
N_j(e_0)*(N_k(e_0)*v)) \, e_0^* \\ &=\ N_j(e_0) * (N_k(e_0) *v).
\end{split}
\end{equation*}
Thus, $(V,*, e_0)$ is a graded, commutative, associative algebra
with unit $e_0$. It is straightforward to check that $\CB$ is
compatible with the product.
\end{proof}
\section{Quantum Products}
\label{sec:quantum}
By a quantum product we will mean a suitable deformation of the
(constant) product on a graded, polarized, real Frobenius algebra.
The weight-three case has been extensively studied in the context of
mirror symmetry for Calabi-Yau threefolds (\cite[Chapter
8]{bo:CK-mirror}, \cite{ar:greg-higgs}). Here we will restrict our
attention to the $k=4$ case. In order to motivate our definitions, we
recall the construction of the Gromov-Witten potential in the case of
Calabi-Yau fourfolds; we refer to \cite[Ch. 7 and 8]{bo:CK-mirror} for
proofs and details.
As in Example~\ref{ex:classical_potential_4}, let $X$ be a Calabi-Yau
fourfold, and consider the graded, polarized real Frobenius algebra $V
= \oplus_{p=0}^4 H^{p,p}(X)$ , endowed with the cup product and the
intersection form $\CB$. We choose a basis $\{T_0,\dots,T_m\}$ as in
the example with the added assumption that $\{T_{1}, \ldots, T_{r}\}$
be a $\Z$-basis of $H^{1,1}(X,\Z)$ lying in the closure of the
K\"ahler cone.
Following~\cite[\S~8.2]{bo:CK-mirror}, we define the Gromov-Witten
potential as
\begin{equation}\label{gromovwitten}
\phi(z) = \phi^{GW}(z) = \sum_n \sum_{\beta\in H_2(X,\Z)}
\frac{1}{n!} \langle I_{0,n,\beta}\rangle (\gamma^n)q^\beta
\end{equation}
where $\gamma = \sum_{j=0}^m z_j T_j$ and $\langle
I_{0,n,\beta}\rangle$ is the Gromov-Witten invariant
\cite[(7.11)]{bo:CK-mirror}. The term $q^\beta$ may be interpreted as
a formal power or, given a class $\omega$ in the complexified K\"ahler
cone, as $q^\beta := \exp(2\pi i \int_\beta \omega)$.
The term corresponding to $\beta = 0$ in (\ref{gromovwitten}) yields
the classical potential ${\phi}_0^{cup}(z) = (1/6)\,\int_X \gamma^3$;
moreover, if we set $\delta = \sum_{j=1}^r z_{j} T_{j}$ and $\epsilon
= \gamma - \delta - z_0 T_0$ and apply the Divisor Axiom (see~\cite[\S
8.3.1]{bo:CK-mirror}) we may rewrite (\ref{gromovwitten}) as
\begin{equation*}
\phi^{GW}(z) = {\phi}_0^{cup}(z) + \sum_n \sum_{\beta\in
H_2(X,\Z)-\{0\}} \frac{1}{n!} \langle I_{0,n,\beta}\rangle
(\epsilon^n) \exp(\int_\beta \delta) q^\beta.
\end{equation*}
Now, the homogeneity properties of the Gromov-Witten potential allow
us to further simplify this expression in case $X$ is a Calabi-Yau
fourfold
\begin{equation*}
\phi^{GW}(z) = \phi_0^{cup}(z) + \sum_{a=1}^s \sum_{\beta\in
H_2(X,\Z)-\{0\}} \langle I_{0,1,\beta}\rangle
(T_{r+a}) z_{r+a} e^{2\pi i \sum_{j=1}^r z_{j} \int_\beta T_{j}}
q^\beta.
\end{equation*}
Note that the above series depends linearly on $z_{r+1},\dots,z_{r+s}$
while the variables $z_{1},\dots,z_{r}$, appear only in exponential
form. Hence, we can write
\begin{equation*}
\phi^{GW}(z) = \phi_0^{cup}(z) + \sum_{a=1}^s z_{r+a}
\phi_h^a(z_{1},\ldots,z_{r}).
\end{equation*}
with $\phi_h^a(z_{1},\ldots,z_{r}) = \Psi^a(e^{2\pi i
z_{1}},\ldots,e^{2\pi i z_{r}})$. It follows from the Effectivity
Axiom (see~\cite[\S~7.3]{bo:CK-mirror}) that $\Psi^a(0) = 0$.
This construction motivates the following definition of an abstract
potential function for graded, polarized, real Frobenius algebras of
weight four.
\begin{definition}\label{quantumpotential}
Let $(V,*_0,e_0,\CB)$ be a graded, polarized, real Frobenius algebra
of weight four and let $\{T_{0},\dots,T_{m}\}$ be an adapted basis as
in Example~\ref{ex:classical_potential_4}. Assume, moreover, that
$T_1,\dots,T_r$ are a framing of $V$. A \textsl{potential} on
$(V,*_0,e_0,\CB)$ is a function
\begin{equation}\label{generalpotential}
\phi(z)=\phi_0(z) + \phi_\hbar(z) \ ;\quad \phi_\hbar(z) =
\sum_{a=1}^s z_{r+a}\, \phi_h^a(z_{1},\ldots,z_{r})\,,
\end{equation}
where $\phi_0(z)$ is the classical potential associated with
$(V,*_0,e_0,\CB)$, $\phi_h^a(z_{1},\ldots,z_{r}) =
\psi^a(q_1,\dots,q_r)$, $q_j =\exp(2\pi i z_{j})$, and $\psi^a(q)$
are holomorphic functions in a neighborhood of the origin in $\C^r$
such that $\psi^a(0)=0$. We will refer to $\phi_\hbar(z)$ as the
{\sl quantum} part of the potential.
\end{definition}
Given a potential $\phi$ we define a {\sl quantum} product on $V$ by
\begin{equation}\label{quantumproduct}
T_a * T_b \ =\ \sum_{c=0}^m \frac{\del^3 \phi}{\del z_a \del z_b
\del z_c} T_c^\CB\,,
\end{equation}
where $\{T_0^\CB,\dots,T_m^\CB\}$ denotes the $\CB$-dual basis.
Clearly, $*$ is commutative and $e_0 = T_0$ is still a unit. The
quantum product is associative if and only if the potential satisfies
the WDVV equations:
\begin{equation*}
\sum_{a=1}^s \frac{\del^3 \phi}{\del z_{i} \del z_{j} \del z_{r+a}}
\ \frac{\del^3 \phi}{\del z_{k} \del z_{l} \del z_{r+a}} \
=\ \sum_{a=1}^s \frac{\del^3 \phi}{\del z_{k} \del z_{j} \del
z_{r+a}} \ \frac{\del^3 \phi}{\del z_{i} \del z_{l} \del
z_{r+a}},
\end{equation*}
with $i,j,k,l$ running from $1$ to $r$. In view of
(\ref{generalpotential}) and (\ref{eq:classical_potential_4}) these
equations are equivalent to (\ref{eq:prod_pot_2}) and
\begin{multline}\label{eq:WDVV2}
\sum_{a=1}^s (P^a_{ij}\frac{\del^2 \phi_h^a}{\del z_{k} \del z_{l}} +
P^a_{kl} \frac{\del^2 \phi_h^a}{\del z_{i} \del z_{j}} +
\frac{\del^2 \phi_h^a}{\del z_{i} \del z_{j}} \frac{\del^2
\phi_h^a}{\del z_{k} \del z_{l}}) = \\= \sum_{a=1}^s
(P^a_{il}\frac{\del^2 \phi_h^a}{\del z_{j} \del z_{k}} + P^a_{jk}
\frac{\del^2 \phi_h^a}{\del z_{i} \del z_{l}} + \frac{\del^2
\phi_h^a}{\del z_{i} \del z_{l}}\frac{\del^2 \phi_h^a}{\del z_{j}
\del z_{k}}),
\end{multline}
for all $i,j,k,l=1,\dots,r$, and where $P^a_{ij}$ denotes the
coefficients (\ref{polynomials}).
\begin{remark} For a classical potential the WDVV equations reduce to the
algebraic relation (\ref{eq:prod_pot_2}). The Gromov-Witten
potential (\ref{gromovwitten}) satisfies the WDVV equations
(see~\cite[Theorem 8.2.4]{bo:CK-mirror}).
\end{remark}
We can now state and prove the main theorem of this section.
\begin{theorem}\label{main}
There is a one-to-one correspondence between
\begin{itemize}
\item Associative quantum products on a framed Frobenius algebra of
weight four.
\item Germs of polarized variations of Hodge structure of weight
four for which the origin $0\in \C^r$ is a maximally unipotent
boundary point, and whose limiting mixed Hodge structure is of
Hodge-Tate type.
\end{itemize}
This correspondence, which depends on the choice of an element
corresponding to the unit, identifies the classical potential with
the nilpotent orbit of the PVHS while the quantum part of the
potential is equivalent to the holomorphic function $\Gamma$ defined
by (\ref{gamma}) relative to a canonical basis.
\end{theorem}
\begin{proof}
Let $(V,*_0,\CB,e_0)$ be a graded, polarized, real Frobenius algebra
of weight four. Let $T_0,\dots,T_m$ be an adapted basis such that
$T_1,\dots,T_r$ is a framing of $V_2$. Let $\HH\subset V_2$ be the
tube domain
\begin{equation*}
\HH \ :=\ \{\,\sum_{j=1}^r \,z_j\,T_j\ ;\ {\rm Im}(z_j) > 0\,\}
\end{equation*}
We view $\HH$ as the universal covering of $(\Delta^*)^r$ via the
map $(z_1,\dots,z_r) \mapsto (q_1,\dots,q_r)$, $q_j = \exp(2\pi i
z_j)$. Let $\VV$ denote the trivial bundle over $(\Delta^*)^r$ with
fiber $V$ and $\FF^p$ the trivial subbundle with fiber $\sum_{a \leq
8-2p} V_{a}$.
Given a potential $\phi$ and elements $w\in V_2$, $v\in V$, the
quantum product $w*v$ may be thought of as a $V$-valued function on
$V$. Let $w *_s v$ denote its restriction to $\HH$. It follows
easily from (\ref{generalpotential}) and
(\ref{eq:classical_potential_4}) that $w *_s v$ depends only on
$q_1,\dots,q_r$ and, therefore, it descends to a $V$-valued function
on $(\Delta^*)^r$, \ie\ a section of $\VV$. This allows us to
define a connection $\nabla$ on $\VV$ by
\begin{equation*}
\nabla_\pd{}{q_j} v \ := \ \frac 1 {2\pi i q_j} (T_{j} *_s v)\,,
\end{equation*}
where in the left-hand side $v$ represents the constant section
defined by $v\in V$. As shown in \cite[Proposition
8.5.2]{bo:CK-mirror} the WDVV equations for the potential $\phi$
imply that $\nabla$ is flat.
We can compute explicitly the connection forms relative to the
constant frame $\{T_0,\dots,T_m\}$. We have, for $j,l=1,\ldots,r$
and $a=1,\ldots,s$:
\begin{equation}\label{eq:connection}
\begin{split}
\nabla_{\pd{}{q_j}} T_0 &= \frac{1}{2\pi i q_j} T_{j}\\
\nabla_{\pd{}{q_j}} T_{l} &= \frac{1}{2\pi i} \sum_{b=1}^s
\left( \frac{P^b_{jl}}{q_j} + 2\pi i \pd{}{q_j}(2\pi i q_l
\pd{\psi^b}{q_l})\right) T_{r+b}\\ \nabla_{\pd{}{q_j}} T_{r+a}
&= \frac{1}{2\pi i} \sum_{k=1}^r \left(\frac{P^a_{jk}}{q_j} +
2\pi i \pd{}{q_j}(2\pi i q_k \pd{\psi^a}{q_k})\right)
T_{r+s+k}\\ \nabla_{\pd{}{q_j}} T_{r+s+l} &= \frac{1}{2\pi i
q_j} \delta_{jl} T_m\\ \nabla_{\pd{}{q_j}} T_m &= 0,
\end{split}
\end{equation}
where the coefficients $P^a_{jk}$ are defined as in
(\ref{polynomials}). Note that these equations imply that the
bundles $\FF^p$ satisfy the horizontality condition
(\ref{horizontality}). Moreover, suppose we define a bilinear form
$Q$ on $V$ as in (\ref{eq:Q_from_B}) and extend it trivially to a
form $\QQ$ on $\VV$, then it is straightforward to check that
\begin{equation*}
\QQ( \nabla_{\pd{}{q_j}} T_a , T_b) + \QQ( T_a ,
\nabla_{\pd{}{q_j}}T_b) =0
\end{equation*}
for all $j=1,\dots,r$ and all $a,b = 0,\dots,m$. Hence the form
$\QQ$ is $\nabla$-flat.
We may also deduce from (\ref{eq:connection}) that $\nabla$ has a
simple pole at the origin with residues
\begin{equation*}
\res_{q_j=0}(\nabla) \ =\ \frac{1}{2\pi i}\ L^0_{T_j}\ ;\quad
j=1,\dots,r,
\end{equation*}
where $L^0_{T_j}$ denotes multiplication by $T_j$ relative to the
constant product $*_0$. It then follows from
\cite[Th\'eor\`eme~II.1.17]{bo:Del-equations} that, written in terms
of the (multivalued) $\nabla$-flat basis $T_0^\flat,\dots,T_m^\flat$
the matrix of the monodromy logarithms $N_j$ coincides with the
matrix of $2\pi i \res_{q_j=0}(\nabla)$ in the constant basis
$T_0,\dots,T_m$, \ie\ with $L^0_{T_j}$. Because the operators
$L^0_{T_j}$ are real, so is the monodromy $\exp(N_j)$ and therefore
we can define a flat real structure $\VV_\R$ on $\VV$.
Since $T_1,\dots,T_r$ are a framing of the polarized Frobenius
algebra $(V,*_0,\CB,e_0)$, it follows from (\ref{theta}) that the map
\begin{equation*}
\theta(q_1,\dots,q_r) \ =\ \exp(\sum_{j=1}^r z_j N_{j}) \cdot F\
;\quad z_j = e^{2\pi i q_j}\,.
\end{equation*}
is the period map of a VHS (a nilpotent orbit) in the bundle
$(\VV,\VV_\R,\nabla,\QQ)$. Since the bundles $\FF^p$ are already
known to satisfy (\ref{horizontality}), we can apply
Theorem~\ref{th:2.8} to conclude that they define a polarized VHS on
$(\VV,\VV_\R,\nabla,\QQ)$.
In order to complete the asymptotic description of the PVHS defined
by $\FF$ on $\VV$, we need to compute the holomorphic function
$\Gamma \colon \Delta^r \to \jlg_-$. Because of
Theorem~\ref{th:improved_2.8}, it suffices to determine the
component $\Gamma_{-1}$. Moreover, it follows from
Proposition~\ref{prop:special_coordinates} that we may choose
canonical coordinates $(q_1,\dots,q_r)$ on $\Delta^r$, so that, in
terms of the basis $T_0,\dots,T_m$, $\Gamma(q)$ has the form:
\begin{equation}\label{gammacan}
\Gamma(q) =
\left(
\begin{array}{c|c|c|c|c}
& & & & \\\hline
0 & & & & \\\hline
- \transpose{D}(q) & \transpose{C}(q) & & &\\\hline
*& * & C(q)& &\\\hline
* & * & D(q) & 0&
\end{array}
\right).
\end{equation}
Thus, $\Gamma(q)$ is completely determined by the $r\times s$-matrix
$C(q)$.
On the other hand, as noted in \S \ref{sec:variations_at_infinity},
$\Psi(q) = \exp \Gamma(q)\cdot F_0$ is the expression of the Hodge
bundles $\FF^p$ in terms of the canonical sections
(\ref{cansections}):
\begin{equation}\label{cansections2}
\tilde T(z) \ =\
\exp (\sum_{j=1}^r\, z_j N_j)\cdot T^{\flat}\,
\end{equation}
The matrix $\exp (-\Gamma(q))$, in the basis $T_0,\dots,T_m$, is the
matrix expressing the canonical sections $\tilde T_0,\dots,\tilde
T_m$ in terms of the constant frame. Therefore
\begin{equation*}
\tilde T_{r+a}(q) \ =\ T_{r+a} - \sum_{k=1}^r C_{ka}(q) T_{r+s+k}
- D_a(q)T_m
\end{equation*}
and it suffices to compute $\tilde T_{r+a}(q)$ to determine $C$ (and
$D$).
It is straightforward to show, using the formulae
(\ref{eq:connection}), that
\begin{equation*}
\begin{split}
T_m^\flat &= T_m\\ T_{r+s+l}^\flat &= T_{r+s+l} - z_l T_m\\
T_{r+a}^\flat &=T_{r+a} -\sum_{l=1}^r \pd{(P^a + \phi_h^a)}{z_l}
T_{r+s+l} + (P^a + \phi_h^a) T_m
\end{split}
\end{equation*}
Hence, to obtain $\tilde T_{r+a}(q)$ it suffices to apply
(\ref{cansections2}), together with the fact that the matrix of
$N_j$ in the basis $\{T_p^\flat\}$ coincides with that of
$*_0$-multiplication by $T_j$ relative to $\{T_p\}$. Thus,
$N_j(T_m^\flat) = 0$ and
\begin{equation*}
N_j(T_{r+s+l}^\flat)\ =\ \delta_{jl}\, T_m^\flat\,;\quad
N_j(T_{r+a}^\flat)\ =\ \sum_{k=1}^r P^a_{jk} \,T_{r+s+k}^\flat\,.
\end{equation*}
This, together with the fact that $P^a$ is a homogeneous polynomial
of degree $2$, implies that
\begin{equation*}
\begin{split}
(\sum_{j=1}^r z_j N_j) T^\flat_{r+a} &= \sum_{j,l=1}^r z_j
\frac{\del^2 P^a}{\del z_j \del z_l} T^\flat_{r+s+l} =
\sum_{l=1}^r \pd{P^a}{z_l} T^\flat_{r+s+l} \\ &= \sum_{l=1}^r
\pd{P^a}{z_l} (T_{r+s+l} - z_l T_m) = \sum_{l=1}^r \pd{P^a}{z_l}
T_{r+s+l} - 2 P^a T_m
\end{split}
\end{equation*}
and
\begin{equation*}
\frac{1}{2}(\sum_{j=1}^r z_j N_j)^2 T^\flat_{r+a} =
\frac{1}{2}(\sum_{j=1}^r z_j N_j) \sum_{l=1}^r \pd{P^a}{z_l}
T^\flat_{r+s+l} = \frac{1}{2}\sum_{j=1}^r z_j \pd{P^a}{z_j}
T^\flat_m = P^a T_m.
\end{equation*}
Hence
\begin{equation*}
\begin{split}
\ti{T}_{r+a}\ &=\ T^\flat_{r+a} + (\sum_{j=1}^r z_j N_j)
T^\flat_{r+a} + \frac{1}{2} (\sum_{j=1}^r z_j N_j)^2
T^\flat_{r+a}\\ &=\ T^\flat_{r+a} + \sum_{l=1}^r \pd{P^a}{z_l}
T_{r+s+l} - 2 P^a T_m + P^a T_m \\ &=\ T_{r+a} - \sum_{l=1}^r
\pd{\phi_h^a}{z_l} T_{r+s+l} + \phi_h^a T_m.
\end{split}
\end{equation*}
Thus,
\begin{equation*}
C_{ka}\ =\ \pd{\phi_h^a}{z_k}\;\quad \hbox{and}\quad
D_a\ = \ -\phi_h^a\,.
\end{equation*}
Conversely, suppose now that $(\VV,\VV_\R,\FF,\nabla,\QQ)$ is a
polarized VHS of weight four over $(\Delta^*)^r$, that the origin is
a maximally unipotent boundary point, and that the limiting MHS is
of Hodge-Tate type. Let $\{N_1,\dots,N_r;F\}$ denote the associated
nilpotent orbit and set $V_{8-2p} := I^{p,p}$. It follows from
Proposition~\ref{prop:constant_prod_from_nilpotent} that we can
define a product $*_0$, and a bilinear form $\CB$ ---as in
(\ref{eq:Q_from_B})--- turning $(V,*_0,\CB)$ into a polarized, real,
graded Frobenius algebra with unit $e_0\in V_0$. This structure is
determined by the choice of unit and the fact that, relative to an
adapted basis $\{T_0,\dots,T_m\}$ as in
Example~\ref{ex:classical_potential_4},
\begin{equation}\label{coeff}
N_j (T_{r+a}) \ =\ \sum_{k=1}^r \CB(T_j*_0 T_{r+a}, T_k)\,T_{r+s+k}
\ =\ \sum_{k=1}^r P^a_{jk}\,T_{r+s+k},
\end{equation}
$j=1,\dots,r$, $a=1,\dots,s$, and $P^a_{jk}$ are the coefficients
(\ref{polynomials}) of the associated classical potential $\phi_0$.
We have already noted that in canonical coordinates, the holomorphic
function $\Gamma$ associated with the PVHS takes on the special form
(\ref{gammacan}). Moreover, since $\Gamma(q)$ satisfies the
differential equation (\ref{eq:integcond}), we have from
(\ref{eq:e7}) that
\begin{equation*}
dG_{-2} = \Gamma_{-1} \Theta - \Theta \Gamma_{-1} + \Gamma_{-1}
d\Gamma_{-1}
\end{equation*}
for $\Theta = d(\sum_{j=1}^r z_{j} {N}_j)$. Consequently,
\begin{equation}\label{int2}
d(D_a) = - \sum_{k=1}^r C_{ka}\,dz_k.
\end{equation}
We now define a potential on $V$ by
\begin{equation}\label{pot}
\phi(z)\ :=\ \phi_0(z) - \sum_{a=1}^s z_{r+a} D_a(q)\,.
\end{equation}
Since we already know that the classical potential $\phi_0$
satisfies (\ref{eq:prod_pot_2}), the WDVV equations for $\Phi$
reduce to the equations (\ref{eq:WDVV2}). But this is a consequence
of the integrability condition (\ref{eq:integcond}); indeed, note
that given (\ref{coeff}), if we let $\Xi = (\xi_{ka})$ be the
$r\times s$-matrix of one forms
\begin{equation*}
\xi_{ka}\ =\ \sum_{j=1}^r (P^a_{jk} + \pd {C_{ka}}{z_j})\,dz_j,
\end{equation*}
the equation (\ref{eq:integcond}) reduces to
\begin{equation}
\label{eq:WDVV3}
\Xi \wedge \Xi^t \ =\ 0
\end{equation}
which, in view of (\ref{int2}) and (\ref{pot}), is easily seen to be
equivalent to (\ref{eq:WDVV2}).
\end{proof}
\begin{remark}
Note that~(\ref{eq:WDVV3}) expresses the WDVV equations in a very
compact form. Also, note that even though the quantum product is
defined in terms of third derivatives of $\phi$, one recovers the
full potential $\phi$ from the PVHS. Since~(\ref{quantumproduct})
only allows for an ambiguity which is, at most, quadratic in $z$,
the quantum part $\phi_\hbar$ is uniquely determined.
\end{remark}
\newpage
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
| 113,634
|
\draftcut
\section{w-Braids} \label{sec:w-braids}
\begin{quote} \small {\bf Section Summary. }
\summarybraids
\end{quote}
\subsection{Preliminary: Virtual Braids, or v-Braids.}
\label{subsec:VirtualBraids}
Our main object of study for this section, w-braids, are best
viewed as ``virtual braids''~\cite{Bardakov:VirtualAndUniversal,
KauffmanLambropoulou:VirtualBraids, BardakovBellingeri:VirtualBraids},
or v-braids, modulo one additional relation; hence, we start with v-braids.
It is simplest to define v-braids in
terms of generators and relations, either algebraically or pictorially.
This can be done in at least two ways -- the easier-at-first but
philosophically less satisfying ``planar'' way, and the harder-to-digest
but morally more correct ``abstract'' way.\footnote{Compare with a
similar choice that exists in the definition of manifolds, as either
appropriate subsets of some ambient Euclidean spaces (module some
equivalences) or as abstract gluings of coordinate patches (modulo some
other equivalences). Here in the ``planar'' approach of
Section~\ref{subsubsec:Planar} we consider v-braids
as ``planar'' objects, and in the ``abstract approach'' of
Section~\ref{subsubsec:Abstract} they are just ``gluings'' of abstract
``crossings'', not drawn anywhere in particular.}
\subsubsection{The ``Planar'' Way} \label{subsubsec:Planar} For a
natural number $n$ set $\glos{\vB_n}$ to be the group generated by
symbols $\glos{\sigma_i}$ ($1\leq i\leq n-1$), called ``crossings''
and graphically represented by an overcrossing $\overcrossing$ ``between
strand $i$ and strand $i+1$'' (with inverse $\undercrossing$)\footnote{We
sometimes refer to $\overcrossing$ as a ``positive crossing'' and to
$\undercrossing$ as a ``negative crossing''.}, and $\glos{s_i}$, called
``virtual crossings'' and graphically represented by a non-crossing,
$\virtualcrossing$, also ``between strand $i$ and strand $i+1$'',
subject to the following relations:
\begin{myitemize}
\item The subgroup of $\vB_n$ generated by the virtual crossings $s_i$
is the symmetric group $\glos{S_n}$, and the $s_i$'s correspond to the
transpositions $(i,i+1)$. That is, we have
\begin{equation} \label{eq:sRelations}
s_i^2=1,
\qquad s_is_{i+1}s_i = s_{i+1}s_is_{i+1},
\qquad\text{and if $|i-j|>1$, then}
\qquad s_is_j=s_js_i.
\end{equation}
In pictures, this is
\begin{equation} \label{eq:sRels}
\def\i{{$i$}}
\def\ip{{$i\!+\!1$}}
\def\ipp{{$i\!+\!2$}}
\def\j{{$j$}}
\def\jp{{$j\!+\!1$}}
\pstex{sRels}
\end{equation}
Note that we read our braids from bottom to top, and that all relations (and most pitcures in this paper) are local:
the braids may be bigger than shown but the parts not shown remain the same throughout a relation.
\item The subgroup of $\vB_n$ generated by the crossings $\sigma_i$'s is
the usual braid group $\glos{\uB_n}$, and $\sigma_i$ corresponds to the ``braiding
of strand $i$ over strand $i+1$''. That is, we have
\begin{equation} \label{eq:R3}
\sigma_i\sigma_{i+1}\sigma_i
= \sigma_{i+1}\sigma_i\sigma_{i+1},
\qquad\text{and if $|i-j|>1$ then}
\qquad \sigma_i\sigma_j=\sigma_j\sigma_i.
\end{equation}
In pictures, dropping the indices, this is
\begin{equation} \label{eq:sigmaRels} \pstex{sigmaRels} \end{equation}
The first of these relations is the ``Reidemeister 3 move''\footnote{The
Reidemeister 2 move is the relations $\sigma_i\sigma_i^{-1}=1$ which is
part of the definition of a group. There is no Reidemeister 1 move in
the theory of braids.} of knot theory. The second is sometimes called
``locality in space''~\cite{Bar-Natan:NAT}.
\item Some ``mixed relations'', that is,
\begin{equation} \label{eq:MixedRelations}
s_i\sigma^{\pm 1}_{i+1}s_i = s_{i+1}\sigma^{\pm 1}_is_{i+1},
\qquad\text{and if $|i-j|>1$, then}
\qquad s_i\sigma_j=\sigma_js_i.
\end{equation}
In pictures, this is
\begin{equation} \label{eq:MixedRels}
\pstex{MixedRels}
\end{equation}
\end{myitemize}
\begin{remark} \label{rem:Skeleton} The ``skeleton'' of a v-braid $B$
is the set of strands appearing in it, retaining the association
between their beginning and ends but ignoring all the crossing
information. More precisely, it is the permutation induced by
tracing along $B$, and even more precisely it is the image of $B$
via the ``skeleton morphism'' $\glos{\varsigma}\colon\vB_n\to S_n$ defined
by $\varsigma(\sigma_i)=\varsigma(s_i)=s_i$ (or pictorially, by
$\varsigma(\overcrossing)=\varsigma(\virtualcrossing)=\virtualcrossing$).
Thus, the symmetric group $S_n$ is both a subgroup and a quotient group
of $\vB_n$.
\end{remark}
Like there are pure braids to accompany braids, there are pure virtual
braids as well:
\begin{definition} A pure v-braid is a v-braid whose skeleton is the
identity permutation; the group $\glos{\PvB_n}$ of all pure v-braids is
simply the kernel of the skeleton morphism $\varsigma\colon\vB_n\to S_n$.
\end{definition}
We note the short exact sequence of group homomorphisms
\begin{equation} \label{eq:ExcatSeqForPvB}
1\longrightarrow\PvB_n\xhookrightarrow{\quad}\vB_n
\overset{\varsigma}{\longrightarrow}S_n
\longrightarrow 1.
\end{equation}
This short exact sequence splits, with the splitting given by the inclusion
$S_n\hookrightarrow\vB_n$ mentioned above~\eqref{eq:sRelations}.
Therefore, we have that
\begin{equation} \label{eq:vBSemiDirect}
\vB_n=\PvB_n\rtimes S_n.
\end{equation}
\subsubsection{The ``Abstract'' Way} \label{subsubsec:Abstract}
The relations~\eqref{eq:sRels} and~\eqref{eq:MixedRels} that govern
the behaviour of virtual crossings precisely say that virtual crossings
really are ``virtual'' --- if a piece of strand is routed within a braid
so that there are only virtual crossings around it, it can be rerouted
in any other ``virtual only'' way, provided the ends remain fixed
(this is Kauffman's ``detour move''~\cite{Kauffman:VirtualKnotTheory,
KauffmanLambropoulou:VirtualBraids}). Since a v-braid $B$ is independent
of the routing of virtual pieces of strand, we may as well never supply
this routing information.
\parpic[r]{$\pstex{PvBExample}$}
Thus, for example, a perfectly fair verbal description of the
(pure!) v-braid on the right is ``strand 1 goes over strand 3 by a
positive crossing then likewise positively over strand 2 then negatively
over 3 then 2 goes positively over 1''. We don't need to specify how
strand 1 got to be near strand 3 so it can go over it --- it got there
by means of virtual crossings, and it doesn't matter how. Hence we arrive
at the following ``abstract'' presentation of $\PvB_n$ and $\vB_n$:
\begin{proposition} (E.g.~\cite[Theorems 1 and 2]{Bardakov:VirtualAndUniversal})
\begin{enumerate}
\item The group $\PvB_n$ of pure v-braids
is isomorphic to the group generated by symbols $\glos{\sigma_{ij}}$ for
$1\leq i\neq j\leq n$ (meaning ``strand $i$ crosses over strand $j$
at a positive crossing''\footnote{The inverse, $\sigma_{ij}^{-1}$,
is ``strand $i$ crosses over strand $j$ at a negative crossing''}),
subject to the third Reidemeister move and to locality in space (compare
with~\eqref{eq:R3} and~\eqref{eq:sigmaRels}):
\begin{align*}
\sigma_{ij}\sigma_{ik}\sigma_{jk} &= \sigma_{jk}\sigma_{ik}\sigma_{ij}
& \text{whenever}\qquad & |\{i,j,k\}|=3, \\
\sigma_{ij}\sigma_{kl} &= \sigma_{kl}\sigma_{ij}
& \text{whenever}\qquad & |\{i,j,k,l\}|=4.
\end{align*}
\item If $\tau\in S_n$, then with the action
$\sigma_{ij}^\tau:=\sigma_{\tau i,\tau j}$ we recover the semi-direct
product decomposition $\vB_n=\PvB_n\rtimes S_n$. \qed
\end{enumerate}
\end{proposition}
\draftcut \subsection{On to w-Braids} \label{subsec:wBraids}
To define w-braids, we break the symmetry between overcrossings and
undercrossings by imposing one of the ``forbidden moves'' in virtual knot
theory, but not the other:
\begin{equation} \label{eq:OvercrossingsCommute}
\sigma_i\sigma_{i+1}s_i = s_{i+1}\sigma_i\sigma_{i+1},
\qquad\text{yet}\qquad
s_i\sigma_{i+1}\sigma_i \neq \sigma_{i+1}\sigma_is_{i+1}.
\end{equation}
Alternatively,
\[ \sigma_{ij}\sigma_{ik} = \sigma_{ik}\sigma_{ij},
\qquad\text{yet}\qquad
\sigma_{ik}\sigma_{jk} \neq \sigma_{jk}\sigma_{ik}.
\]
In pictures, this is
\begin{equation} \label{eq:OC} \pstex{OCUC} \end{equation}
The relation we have just imposed may be called the ``unforbidden
relation'', or, perhaps more appropriately, the ``overcrossings commute''
relation, abbreviated \glost{OC}. Ignoring the non-crossings\footnote{Why this is
appropriate was explained in the previous section.} $\virtualcrossing$,
the OC relation says that it is the same if strand $i$ first crosses
over strand $j$ and then over strand $k$, or if it first crosses over
strand $k$ and then over strand $j$. The ``undercrossings commute''
relation \glost{UC}, the one we do not impose
in~\eqref{eq:OvercrossingsCommute}, would say the same except with
``under'' replacing ``over''.
\begin{definition} The group of w-braids is $\glos{\wB_n}:=\vB_n/OC$. Note
that $\varsigma$ descends to $\wB_n$, and hence we can define the group $\glos{\PwB_n}$ of
pure w-braids to be the kernel of the map $\varsigma\colon\wB_n\to S_n$. We
still have a split exact sequence as at~\eqref{eq:ExcatSeqForPvB} and
a thus, a semi-direct product decomposition $\wB_n=\PwB_n\rtimes S_n$.
\end{definition}
\begin{exercise} Show that the OC relation is equivalent to the relation
\[
\sigma_i^{-1}s_{i+1}\sigma_i = \sigma_{i+1}s_i\sigma_{i+1}^{-1}
\qquad\text{or}\qquad
\parbox[m]{1.5in}{\input figs/AltOC.pstex_t }
\]
\end{exercise}
While for most of this paper the pictorial / algebraic definition of w-braids
(and other w-knotted objects) will suffice, we ought describe at least
briefly a few further interpretations of $\wB_n$:
\subsubsection{The Group of Flying Rings} \label{subsubsec:FlyingRings}
Let \glos{X_n} be the space of all placements of $n$ numbered disjoint
geometric circles in $\bbR^3$, such that all circles are parallel to
the $xy$ plane. Such placements will be called horizontal\footnote{
For the group of non-horizontal flying rings see
Section \ref{subsubsec:NonHorRings}.}. A horizontal
placement is determined by the centres in $\bbR^3$ of the $n$ circles
and by $n$ radii, so $\dim X_n=3n+n=4n$. The permutation group $S_n$
acts on $X_n$ by permuting the circles, and one may think of the quotient
$\glos{\tilde{X}_n}:=X_n/S_n$ as the space of all horizontal
placements of $n$ unmarked circles in $\bbR^3$. The fundamental group
$\pi_1(\tilde{X}_n)$ is a group of paths traced by $n$ disjoint horizontal
circles (modulo homotopy), so it is fair to think of it as ``the group
of flying rings''.
\begin{theorem} The group of pure w-braids $\PwB_n$ is isomorphic to the group
of flying rings $\pi_1(X_n)$. The group $\wB_n$ is isomorphic to the group
of unmarked flying rings $\pi_1(\tilde{X}_n)$.
\end{theorem}
For the proof of this theorem,
see~\cite{Goldsmith:MotionGroups, Satoh:RibbonTorusKnots} and
especially~\cite[Proposition~3.3]{BrendleHatcher:RingsAndWickets}. Here
we will contend ourselves with pictures describing the images of the
generators of $\wB_n$ in $\pi_1(\tilde{X}_n)$ and a few comments:
\[ \input figs/FlyingRings.pstex_t \]
Thus, we map the permutation $s_i$ to the movie clip in which ring
number $i$ trades its place with ring number $i+1$ by having the two
flying around each other. This acrobatic feat is performed in $\bbR^3$
and it does not matter if ring number $i$ goes ``above'' or ``below''
or ``left'' or ``right'' of ring number $i+1$ when they trade places,
as all of these possibilities are homotopic. More interestingly, we
map the braiding $\sigma_i$ to the movie clip in which ring $i+1$
shrinks a bit and flies through ring $i$. It is a worthwhile
exercise for the reader to verify that the relations in the definition
of $\wB_n$ become homotopies of movie clips. Of these relations it
is most interesting to see why the ``overcrossings commute'' relation
$\sigma_i\sigma_{i+1}s_i = s_{i+1}\sigma_i\sigma_{i+1}$ holds, yet the
``undercrossings commute'' relation $\sigma^{-1}_i\sigma^{-1}_{i+1}s_i =
s_{i+1}\sigma^{-1}_i\sigma^{-1}_{i+1}$ doesn't.
\begin{exercise}\label{ex:swBn}
To be perfectly precise, we have to
specify the fly-through direction. In our notation, $\sigma_i$ means
that the ring corresponding to the strand going under (in the local picture
for $\sigma_i$) approaches from below the bigger
ring representing the strand going over, then flies through it and exists
above. For $\sigma_i^{-1}$ we are ``playing the movie backwards'', i.e.,
the ring of the strand going under comes from above and exits below the ring
of the ``over'' strand.
Let ``the signed $w$ braid group'', $\swB_n$, be the group
of horizontal flying rings where both fly-through
directions are allowed. This introduces a ``sign'' for
each crossing $\sigma_i$:
\begin{center}
\input figs/FlyingRings2.pstex_t
\end{center}
In other words, $\swB_n$ is generated by $s_i$,
$\sigma_{i+}$ and $\sigma_{i-}$, for $i=1,...,n-1$. Check that in $\swB_n$
$\sigma_{i-}=s_i\sigma_{i+}^{-1}s_i$, and this, along with
the other obvious relations implies $\swB_n \cong \wB_n$.
\end{exercise}
\subsubsection{Certain Ribbon Tubes in $\bbR^4$} \label{subsubsec:ribbon}
With
time as the added dimension, a flying ring in $\bbR^3$ traces a tube
(an annulus) in $\bbR^4$, as shown in the picture below:
\[ \input figs/RibbonTubes.pstex_t \]
Note that we adopt here the drawing conventions of Carter and
Saito~\cite{CarterSaito:KnottedSurfaces} --- we draw surfaces as if they
were projected from $\bbR^4$ to $\bbR^3$, and we cut them open whenever
they are ``hidden'' by something with a higher fourth coordinate.
Note
also that the tubes we get in $\bbR^4$ always bound natural 3D
``solids'' --- their ``insides'', in the pictures above. These solids
are disjoint in the case of $s_i$ and have a very specific kind
of intersection in the case of $\sigma_i$ --- these are transverse
intersections with no triple points, and their inverse images are a
meridional disk on the ``thin'' solid tube and an interior disk on the
``thick'' one. By analogy with the case of ribbon knots and ribbon
singularities in $\bbR^3$ (e.g.~\cite[Chapter V]{Kauffman:OnKnots}) and
following Satoh~\cite{Satoh:RibbonTorusKnots}, we call this kind if
intersections of solids in $\bbR^4$ ``ribbon singularities'' and thus, our
tubes in $\bbR^4$ are always ``ribbon tubes''.
\subsubsection{Basis Conjugating Automorphisms of $F_n$}
\label{subsubsec:McCool}
Let
$\glos{F_n}$ be the free (non-abelian) group with generators
$\glos{\xi_1,\ldots,\xi_n}$. Artin's theorem (Theorems 15 and 16
of~\cite{Artin:TheoryOfBraids}) says that the (usual) braid
group $\uB_n$ (equivalently, the subgroup of $\wB_n$ generated by
the $\sigma_i$'s) has a faithful right action on $F_n$. In other
words, $\uB_n$ is isomorphic to a subgroup $H$ of $\Autop(F_n)$
(the group of automorphisms of $F_n$ with opposite multiplication, i.e.,
$\psi_1\psi_2:=\psi_2\circ\psi_1$). Precisely, using $(\xi,
B)\mapsto\xi\glos{\sslash}B$ to denote the right action of $\Autop(F_n)$ on
$F_n$, the subgroup $H$ consists of those automorphisms $B\colon F_n\to F_n$
of $F_n$ that satisfy the following two conditions:
\begin{enumerate}
\item $B$ maps any generator $\xi_i$ to a
conjugate of a generator (possibly different). That is, there is a
permutation $\beta\in S_n$ and elements $a_i\in F_n$ so that, for every $i$,
\begin{equation} \label{eq:BasisConjugating}
\xi_i \sslash B = a_i^{-1}\xi_{\beta (i)}a_i.
\end{equation}
\item $B$ fixes the ordered product of the generators of $F_n$,
\[ \xi_1\xi_2\cdots \xi_n \sslash B = \xi_1\xi_2\cdots \xi_n. \]
\end{enumerate}
McCool's theorem\footnote{Stricktly speaking, the main theorem
of~\cite{McCool:BasisConjugating} is about $\PwB_n$, yet it can easily be
restated for $\wB_n$.}~\cite{McCool:BasisConjugating} says that almost the same statement
holds true\footnote{Though see Warning~\ref{warn:NoArtin}.} for the bigger group $\wB_n$:
namely, $\wB_n$ is isomorphic to the subgroup of $\Autop(F_n)$ consisting of
automorphisms satisfying only the first condition above.
So $\wB_n$ is precisely the group
of ``basis-conjugating'' automorphisms of the free group $F_n$,
the group of those automorphisms which map any ``basis element''
in $\{\xi_1,\ldots,\xi_n\}$ to a conjugate of a (possibly different)
basis element.
The relevant action is explicitly defined on the generators of $\wB_n$
and $F_n$ as follows (we state how each generator of $\wB_n$ acts on
each generator of $F_n$, in each case omitting the generators of $F_n$
which are fixed under the action):
\begin{equation} \label{eq:ExplicitPsi}
(\xi_i, \xi_{i+1})\sslash s_i = (\xi_{i+1}, \xi_i),
\qquad
(\xi_i, \xi_{i+1})\sslash \sigma_i = (\xi_{i+1},
\xi_{i+1}\xi_i\xi_{i+1}^{-1}),
\qquad
\xi_j\sslash \sigma_{ij} = \xi_i\xi_j\xi_i^{-1}.
\end{equation}
It is a worthwhile exercise to verify that $\sslash$ respects the
relations in the definition of $\wB_n$ and that the permutation $\beta$
in~\eqref{eq:BasisConjugating} is the skeleton $\varsigma(B)$.
There is a more conceptual description of $\sslash$, in terms of the
structure of $\wB_{n+1}$. Consider the inclusions
\begin{equation} \label{eq:inclusions}
\wB_n \xhookrightarrow{\iota} \wB_{n+1} \xhookleftarrow{i_u} F_n.
\end{equation}
Here $\glos{\iota}$ is the inclusion of $\wB_n$ into $\wB_{n+1}$ by adding an
inert $(n+1)$st strand (it is injective as it has a well-defined
one sided inverse -- the deletion of the $(n+1)$st strand).
\parpic[r]{$\pstex{xi}$}
The inclusion $\glos{i_u}$ of the free group $F_n$ into $\wB_{n+1}$ is defined by
$i_u(\xi_i):=\sigma_{i,n+1}$. The image $i_u(F_n)\subset\wB_{n+1}$ is the
set of all w-braids whose first $n$ strands are straight and vertical,
and whose $(n+1)$-st strand wanders among the first $n$ strands mostly
virtually (i.e., mostly using virtual crossings), occasionally slipping
under one of those $n$ strands, but never going over anything.
It is easier to see that
this is indeed injective using the
``flying rings'' picture of Section~\ref{subsubsec:FlyingRings}. The image
$i_u(F_n)\subset\wB_{n+1}$ can be interpreted as the fundamental group
of the complement in $\bbR^3$ of $n$ stationary rings (which is indeed
$F_n$) --- in $i_u(F_n)$ the only ring in motion is the last, and it only
goes under, or ``through'', other rings, so it can be replaced by a point
object whose path is an element of the fundamental group. The injectivity
of $i_u$ follows from this geometric picture.
\parpic[r]{$\pstex{Bgamma}$} \picskip{4}
One may explicitly verify that $i_u(F_n)$ is normalized by $\iota(\wB_n)$
in $\wB_{n+1}$ (that is, the set $i_u(F_n)$ is preserved by conjugation
by elements of $\iota(\wB_n)$). Thus, the following definition (also shown
as a picture on the right) makes sense, for \linebreak $B\in\wB_n\subset\wB_{n+1}$
and for $\gamma\in F_n\subset\wB_{n+1}$:
\begin{equation} \label{eq:ConceptualPsi}
\gamma\sslash B := i_u^{-1}(B^{-1}\gamma B)
\end{equation}
It is a worthwhile exercise to recover the explicit formulae
in~\eqref{eq:ExplicitPsi} from the above definition.
\begin{warning} \label{warn:NoArtin} People familiar with the Artin story for
ordinary braids should be warned that even though $\wB_n$ acts on $F_n$ and
the action is induced from the inclusions in~\eqref{eq:inclusions} in much
of the same way as the Artin action is induced by inclusions $\uB_n
\xhookrightarrow{\iota} \uB_{n+1} \xhookleftarrow{i} F_n$, there are also
some differences, and some further warnings apply:
\begin{myitemize}
\item In the ordinary Artin story, $i(F_n)$ is the set of braids in
$\uB_{n+1}$ whose first $n$ strands are unbraided (that is, whose image in
$\uB_n$ via ``dropping the last strand'' is the identity). This is not true
for w-braids. For w-braids, in $i_u(F_n)$ the last strand always goes
``under'' all other strands (or just virtually crosses them), but never
``over''.
\item Thus, unlike the isomorphism $\PuB_{n+1}\cong \PuB_n\ltimes F_n$,
it is not true that $\PwB_{n+1}$ is isomorphic to $\PwB_n\ltimes F_n$.
\item The OC relation imposed in $\wB$ breaks
the symmetry between overcrossings and undercrossings. Thus, let
$i_o\colon F_n\to\wB_n$ be the ``opposite'' of $i_u$, mapping into braids in
which the last strand is always ``over'' or virtual. Then $i_o$ is not
injective (its image is in fact abelian) and its image is not normalized
by $\iota(\wB_n)$. So there is no ``second'' action of $\wB_n$ on $F_n$
defined using $i_o$.
\item For v-braids, both $i_u$ and $i_o$ are injective and there are two
actions of $\vB_n$ on $F_n$ --- one defined by first projecting into
w-braids, and the other defined by first projecting into v-braids modulo
``undercrossings commute''. Yet v-braids contain more information than
these two actions can see. The ``Kishino'' v-braid below, for example,
is visibly trivial if either overcrossings or undercrossings are made to
commute, yet by computing its Kauffman bracket we know it is non-trivial
as a v-braid~\cite[``The Kishino Braid'']{WKO}:
\[
\pstex{KishinoBraid} \quad \left(\parbox{1.6in}{\footnotesize
The commutator $ab^{-1}a^{-1}b$ of v-braids $a,b$ annihilated by OC/UC,
respectively, with a minor cancellation.
}\right)
\]
\end{myitemize}
\end{warning}
\begin{problem} \label{prob:wCombing}
Are $\PvB_n$ and $\PwB_n$ semi-direct products of free groups?
For $\PuB_n$, this
is the well-known ``combing of braids'' and it follows from $\PuB_n\cong
\PuB_{n-1}\ltimes F_{n-1}$ and induction.
\end{problem}
\begin{remark} \label{rem:GutierrezKrstic}
Note that Guti\'errez and Krsti\'c~\cite{GutierrezKrstic:NormalForms}
have found ``normal forms'' for the elements of $\PwB_n$, yet they
do not decide whether $\PwB_n$ is ``automatic'' in the sense
of~\cite{Epstein:WordProcessing}.
\end{remark}
\draftcut \subsection{Finite Type Invariants of v-Braids and w-Braids}
\label{subsec:FT4Braids}
Just as we had two definitions for v-braids (and thus, for w-braids)
in Section~\ref{subsec:VirtualBraids}, we will give two
equivalent developments of the theory of finite type invariants of
v-braids and w-braids --- a pictorial/topological version in
Section~\ref{subsubsec:FTPictorial}, and a more abstract algebraic version
in Section~\ref{subsubsec:FTAlgebraic}.
\subsubsection{Finite Type Invariants, the Pictorial Approach}
\label{subsubsec:FTPictorial}
In
the standard theory of finite type invariants of knots (also known as
Vassiliev or Goussarov-Vassiliev invariants)~\cite{Vassiliev:CohKnot, Goussarov:nEquivalence,
Bar-Natan:OnVassiliev, Bar-Natan:EMP}
one progresses from the definition of finite type via iterated
differences to chord diagrams and weight systems, to $4T$ (and other)
relations, to the definition of universal finite type invariants,
and beyond. The exact same progression (with different objects
playing similar roles, and sometimes, when yet insufficiently
studied, with the last step or two missing) is also seen in the theories
of finite type invariants of braids~\cite{Bar-Natan:Braids},
3-manifolds~\cite{Ohtsuki:IntegralHomology, LeMurakamiOhtsuki:Universal,
Le:UniversalIHS}, virtual knots~\cite{GoussarovPolyakViro:VirtualKnots,
Polyak:ArrowDiagrams} and of several other classes of objects. We thus
assume that the reader has familiarity with these basic ideas, and we
only indicate briefly how they are implemented in the case of v-braids
and w-braids.
\begin{figure}
\[ \input figs/Dvh1.pstex_t \]
\caption{
On the left, a 3-singular v-braid and its corresponding 3-arrow
diagram. A self-explanatory algebraic notation for this arrow
diagram is $(\glos{a_{12}a_{41}a_{23}},\,3421)$.
Note that we regard arrow diagrams as graph-theoretic
objects, and hence, the two arrow diagrams on the right, whose
underlying graphs are the same, are regarded as equal. In
algebraic notation this means that we always impose the relation
$a_{ij}a_{kl}=a_{kl}a_{ij}$ when the indices $i$, $j$, $k$, and $l$
are all distinct.
} \label{fig:Dvh1}
\end{figure}
Much
like the formula $\doublepoint\to\overcrossing-\undercrossing$ of
the Vassiliev-Goussarov fame, given a v-braid invariant $\glos{V}\colon
\vB_n\to A$ valued in some abelian group $A$, we extend it to
``singular'' v-braids, i.e., braids that contain ``semi-virtual crossings''
like $\glos{\semivirtualover}$ and $\glos{\semivirtualunder}$ using the
formulae $V(\semivirtualover):=V(\overcrossing)-V(\virtualcrossing)$
and $V(\semivirtualunder):=V(\undercrossing)-V(\virtualcrossing)$
(see~\cite{GoussarovPolyakViro:VirtualKnots, Polyak:ArrowDiagrams,
Bar-NatanHalachevaLeungRoukema:v-Dims}). We
say that ``$V$ is of type $m$'' if its extension vanishes on singular
v-braids having more than $m$ semi-virtual crossings. Up to invariants
of lower type, an invariant of type $m$ is determined by its ``weight
system'', which is a functional $W=\glos{W_m}(V)$ defined on ``$m$-singular
v-braids modulo $\overcrossing=\virtualcrossing=\undercrossing$''. Let
us denote the vector space of all formal linear combinations of such
equivalence classes by $\glos{\calG_m}\calD^v_n$. Much as $m$-singular knots
modulo $\overcrossing=\undercrossing$ can be identified with chord
diagrams, the basis elements of $\calG_m\calD^v_n$ can be identified with
pairs $(D,\beta)$, where $D$ is a horizontal arrow diagram and $\beta$
is a ``skeleton permutation'', see Figure~\ref{fig:Dvh1}.
We assemble the spaces $\calG_m\calD^v_n$ together to form a single
graded space, $\glos{\calD^v_n}:=\oplus_{m=0}^\infty\calG_m\calD^v_n$. Note that
throughout this paper, whenever we write an infinite direct sum,
we automatically complete it. Thus, in $\calD^v_n$ we allow infinite sums
with one term in each homogeneous piece $\calG_m\calD^v_n$, in particular,
exponential-like sums will be heavily used.
\begin{figure}
\[ \input figs/6T.pstex_t \]
\[
a_{ij}a_{ik}+a_{ij}a_{jk}+a_{ik}a_{jk}
= a_{ik}a_{ij}+a_{jk}a_{ij}+a_{jk}a_{ik}
\]
\[
\text{or}\qquad
[a_{ij}, a_{ik}] + [a_{ij}, a_{jk}] + [a_{ik}, a_{jk}] = 0
\]
\caption{The $6T$ relation. Standard knot theoretic conventions apply ---
only the relevant parts of each diagram is shown; in reality each diagram
may have further vertical strands and horizontal arrows, provided the
extras are the same in all 6 diagrams. Also, the vertical strands are in no
particular order --- other valid $6T$ relations are obtained when those
strands are permuted in other ways.} \label{fig:6T}
\end{figure}
\begin{figure}
\[ \begin{array}{ccc}
\input figs/TC.pstex_t & \qquad & \input figs/4TArrow.pstex_t \\
a_{ij}a_{ik} = a_{ik}a_{ij} &&
a_{ij}a_{jk} + a_{ik}a_{jk} = a_{jk}a_{ij} + a_{jk}a_{ik} \\
\text{or} \quad [a_{ij}, a_{ik}] = 0 &&
\text{or} \quad [a_{ij} + a_{ik}, a_{jk}] = 0
\end{array} \]
\caption{The TC and the $\protect\aft$ relations.} \label{fig:TCand4T}
\end{figure}
In the standard finite-type theory for knots, weight
systems always satisfy the $4T$ relation, and are therefore
functionals on $\calA:=\calD/4T$. Likewise, in the case of
v-braids, weight systems satisfy the ``$\glos{6T}$ relation''
of~\cite{GoussarovPolyakViro:VirtualKnots, Polyak:ArrowDiagrams,
Bar-NatanHalachevaLeungRoukema:v-Dims},
shown in Figure~\ref{fig:6T}, and are therefore functionals on
$\glos{\calA^v_n}:=\calD^v_n/6T$. In the case of w-braids, the OC
relation~\eqref{eq:OvercrossingsCommute} implies the
``tails commute'' (\glost{TC}) relation on the level of arrow diagrams,
and in the presence of the TC relation, two of the terms in the $6T$
relation drop out, and what remains is the ``$\glos{\aft}$'' relation. These
relations are shown in Figure~\ref{fig:TCand4T}. Thus, weight systems
of finite type invariants of w-braids are linear functionals on
$\glos{\calA^w_n}:=\calD^v_n/TC,\aft$.
The next question that arises is whether we have already found {\em
all} the relations that weight systems always satisfy. More precisely,
given a degree $m$ linear functional on $\calA^v_n=\calD^v_n/6T$ (or
on $\calA^w_n=\calD^v_n/TC,\aft$), is it always the weight system
of some type $m$ invariant $V$ of v-braids (or w-braids)? As in every
other theory of finite type invariants, the answer to this question
is affirmative if and only if there exists a ``universal finite type
invariant'' (or simply, an ``expansion'') of v-braids (or w-braids):
\begin{definition} \label{def:vwbraidexpansion} An expansion
for v-braids (or w-braids) is an invariant $Z\colon \vB_n\to\calA^v_n$
(or $Z\colon \wB_n\to\calA^w_n$) satisfying the following ``universality
condition'':
\begin{itemize}
\item If $B$ is an $m$-singular v-braid (or w-braid) and
$D\in\calG_m\calD^v_n$ is its underlying arrow diagram as in
Figure~\ref{fig:Dvh1}, then
\[ Z(B)=D+(\text{terms of degree\,}>m). \]
\end{itemize}
\end{definition}
Indeed if $Z$ is an expansion and $W\in\calG_m\calA^\star$,\footnote{$\calA^\star$
here denotes either $\calA^v_n$ or $\calA^w_n$, or in fact, any of many
similar spaces that we will discuss later on.} the universality condition
implies that $W\circ Z$ is a finite type invariant whose weight system
is $W$. To go the other way, if $(D_i)$ is a basis of $\calA$ consisting
of homogeneous elements, if $(W_i)$ is the dual basis of $\calA^\star$ and
$(V_i)$ are finite type invariants whose weight systems are the $W_i$'s,
then $Z(B):=\sum_iD_iV_i(B)$ defines an expansion.
In general, constructing a universal finite type invariant is a
hard task. For knots, one uses either the Kontsevich integral or
perturbative Chern-Simons theory (also known as ``configuration
space integrals''~\cite{BottTaubes:SelfLinking} or ``tinker-toy
towers''~\cite{Thurston:IntegralExpressions}) or the rather fancy
algebraic theory of ``Drinfel'd associators'' (a summary of all those
approaches is at~\cite{Bar-NatanStoimenow:Fundamental}). For homology
spheres, this is the ``LMO invariant''~\cite{LeMurakamiOhtsuki:Universal,
Le:UniversalIHS} (also the ``\AA{}rhus
integral''~\cite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}). For
v-braids, we still don't know if an expansion exists. In contrast, as we shall see
below, the construction of an expansion for w-braids is quite easy.
\subsubsection{Finite Type Invariants, the Algebraic Approach}
\label{subsubsec:FTAlgebraic}
For any group $G$, one can form the group algebra ${\mathbb Q}G$ by
allowing formal linear combinations of group elements and extending
multiplication linearly, where $\mathbb Q$ is the field of rational
numbers\footnote{The definitions in this subsection make sense over $\bbZ$
as well, but the main result of the next subsection requires a field of
characteristic $0$. For simplicity of notation we stick with $\bbQ$.}.
The {\it augmentation ideal} is the ideal generated by
differences, or equivalently, the set of linear combinations of group
elements whose coefficients sum to zero:
\[ \glos{\calI} := \left\{\sum_{i=1}^k a_ig_i\colon
a_i \in {\mathbb Q}, g_i \in G, \sum_{i=1}^k a_i=0\right\}.
\]
Powers of the augmentation ideal provide a filtration of the group
algebra. We denote by $\glos{\calA(G)}:= \bigoplus_{m\geq 0} \calI^m/\calI^{m+1}$
the associated graded space corresponding to this filtration.
\begin{definition}\label{def:grpexpansion} An
expansion for the group
$G$ is a map $Z\colon G \to \calA(G)$, such that the linear extension
$Z\colon {\mathbb Q}G \to \calA(G)$ is filtration preserving and
the induced map $$\gr Z\colon (\gr {\mathbb Q}G=\calA(G)) \to (\gr
\calA(G)=\calA(G))$$ is the identity. An equivalent way to phrase this
is that the degree $m$ piece of $Z$ restricted to $\calI^m$ is the
projection onto $\calI^m/\calI^{m+1}$.
\begin{exercise}\label{ex:BraidsAlgApproach}
Verify that for the groups $\PvB_n$ and $\PwB_n$ the m-th power of the
augmentation ideal coincides with the span of all resolutions of
$m$-singular $v$- or $w$-braids (by a resolution we mean the formal
linear combination where each semivirtual crossing is replaced by
the appropriate difference of a virtual and a regular crossing, as in Figure \ref{fig:Dvh1}). Then
check that the notion of expansion defined above is the same as that of
Definition \ref{def:vwbraidexpansion}, restricted to pure braids.
\end{exercise}
Finally, note the functorial nature of the construction above. What we have
described is a functor from the category of groups to the category of graded
algebras, called {\em projectivization}\footnote{We use this name to
distinguish the associated graded with respect to this particular filtration,
which will be a repeating theme in \cite{Bar-NatanDancso:WKO2}.}
$\proj\colon Grp \to GrAlg$, which assigns to each group
$G$ the graded algebra $\calA(G)$. To each homomorphism $\phi\colon G \to H$,
$\proj$ assigns
the induced map $$\gr \phi\colon (\calA(G)=\gr {\mathbb Q}G) \to (\calA(H)= \gr {\mathbb Q}H).$$
\end{definition}
\draftcut \subsection{Expansions for w-Braids}\label{subsec:wBraidExpansion}
The space $\calA^w_n$ of arrow diagrams on $n$ strands is an associative
algebra in an obvious manner: if the permutations underlying two arrow
diagrams are the identity permutations, then we simply juxtapose the diagrams.
Otherwise we ``slide'' arrows through permutations in the obvious manner
--- if $\tau$ is a permutation, we declare that $\tau a_{(\tau i)(\tau
j)}=a_{ij}\tau$. Instead of seeking an expansion $\wB_n\to\calA^w_n$, we
set the bar a little higher and seek a ``homomorphic expansion'':
\begin{definition} \label{def:Universallity} A homomorphic expansion
$Z\colon \wB_n\to\calA^w_n$ is an expansion that carries products in $\wB_n$
to products in $\calA^w_n$.
\end{definition}
To find a homomorphic expansion, we just need to define it
on the generators of $\wB_n$ and verify that it satisfies the
relations defining $\wB_n$ and the universality condition.
Following~\cite[Section~5.3]{BerceanuPapadima:BraidPermutation}
and~\cite[Section~8.1]{AlekseevTorossian:KashiwaraVergne} we set
$Z(\virtualcrossing)=\virtualcrossing$ (that is, a transposition in
$\wB_n$ gets mapped to the same transposition in $\calA^w_n$, adding no
arrows) and $Z(\overcrossing)=\exp(\rightarrowdiagram)\virtualcrossing$.
(Reacall that we work in the degree completion.)
This last formula is important so deserves to be magnified, explained
and replaced by some new notation:
\begin{equation} \label{eq:reservoir}
Z\left(\!\mathsize{\Huge}{\overcrossing}\!\right)\! =
\exp\left(\!\mathsize{\Huge}{\rightarrowdiagram}\!\right)
\cdot\mathsize{\Huge}{\virtualcrossing}
= \pstex{ZIsExp}+\ldots =: \pstex{ArrowReservoir}.
\end{equation}
Thus the new notation $\overset{e^a}{\longrightarrow}$ stands
for an ``exponential reservoir'' of parallel arrows, much like
$e^a=1+a+aa/2+aaa/3!+\ldots$ is a ``reservoir'' of $a$'s. With
the obvious interpretation for $\overset{e^{-a}}{\longrightarrow}$
(that is, the $-$ sign indicates that the terms should have alternating signs,
as in $e^{-a}=1-a+a^2/2-a^3/3!+\ldots$), the second Reidemeister move
$\overcrossing\undercrossing=1$ forces that we set
\[ Z\left(\mathsize{\Huge}{\undercrossing}\right) =
\mathsize{\Huge}{\virtualcrossing}
\cdot\exp\left(-\mathsize{\Huge}{\rightarrowdiagram}\right)
= \pstex{NegReservoir1} = \pstex{NegReservoir2}.
\]
\begin{theorem} \label{thm:RInvariance} The above formulae define
an invariant $Z\colon \wB_n\to\calA^w_n$ (that is, $Z$ satisfies all the
defining relations of $\wB_n$). The resulting $Z$ is a homomorphic
expansion (that is, it satisfies the universality property of
Definition~\ref{def:Universallity}).
\end{theorem}
\begin{proof} Following~\cite{BerceanuPapadima:BraidPermutation,
AlekseevTorossian:KashiwaraVergne}: for the invariance of $Z$, the
only interesting relations to check are the Reidemeister 3 relation
of~\eqref{eq:sigmaRels} and the OC relation
of~\eqref{eq:OC}. For Reidemeister 3, we have
\[ \pstex{R3Left}
= e^{a_{12}}e^{a_{13}}e^{a_{23}}\tau
\overset{1}{=} e^{a_{12}+a_{13}}e^{a_{23}}\tau
\overset{2}{=} e^{a_{12}+a_{13}+a_{23}}\tau,
\]
where $\tau$ is the permutation $321$ and equality 1 holds because
$[a_{12},a_{13}]=0$ by a TC relation and equality 2 holds
because $[a_{12}+a_{13}, a_{23}]=0$ by a $\aft$ relation.
Likewise, again using TC and $\aft$,
\[ \pstex{R3Right}
= e^{a_{23}}e^{a_{13}}e^{a_{12}}\tau
= e^{a_{23}}e^{a_{13}+a_{12}}\tau
= e^{a_{23}+a_{13}+a_{12}}\tau,
\]
and so Reidemeister 3 holds. An even simpler proof using just the TC
relation shows that the OC relation also holds.
Finally, since $Z$ is homomorphic, it is enough to check the universality
property at degree $1$, where it is very easy:
\[ Z\left(\mathsize{\Huge}{\semivirtualover}\right) =
\exp\left(\mathsize{\Huge}{\rightarrowdiagram}\right)
\cdot\mathsize{\Huge}{\virtualcrossing}
- \mathsize{\Huge}{\virtualcrossing}
= \mathsize{\Huge}{\rightarrowdiagram}\cdot\mathsize{\Huge}{\virtualcrossing}
+ (\text{terms of degree\,}>1),
\]
and a similar computation manages the $\semivirtualunder$ case. \qed
\end{proof}
\begin{remark} \label{rem:YangBaxter} Note that the main ingredient
of the above proof was to show that \linebreak $\glos{R}:=Z(\sigma_{12})=e^{a_{12}}$
satisfies the famed Yang-Baxter equation,
\[ R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12}, \]
where $R^{ij}$ means ``place $R$ on strands $i$ and $j$''.
\end{remark}
\draftcut
\subsection{Some Further Comments} \label{subsec:bcomments}
\subsubsection{Compatibility with Braid Operations}
\label{subsubsec:BraidCompatibility}
As with
any new gadget, we would like to know how compatible the expansion
$Z$ of the previous section is with the gadgets we already have; namely,
with various operations that are available on w-braids and with the action
of w-braids on the free group $F_n$, see Section~\ref{subsubsec:McCool}.
\parpic[r]{$\xymatrix{
\wB_n \ar[r]^\theta \ar[d]_Z & \wB_n \ar[d]^Z \\
\calA^w_n \ar[r]_\theta & \calA^w_n
\ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{$Z$ is Compatible with Braid Inversion} \label{par:theta}
Let $\theta$ denote both the ``braid inversion'' operation
$\glos{\theta}\colon \wB_n\to\wB_n$ defined by $B\mapsto B^{-1}$ and
the ``antipode'' anti-automorphism $\theta\colon \calA^w_n\to\calA^w_n$
defined by mapping permutations to their inverses and arrows to their
negatives (that is, $a_{ij}\mapsto-a_{ij}$). Then the diagram on the
right commutes.
\pagebreak[2]
\parpic[r]{$\xymatrix{
\wB_n \ar[r]^<>(0.5)\Delta \ar[d]_Z & \wB_n\times\wB_n \ar[d]^{Z\times Z} \\
\calA^w_n \ar[r]_<>(0.5)\Delta & \calA^w_n\otimes\calA^w_n
\ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Braid Cloning and the Group-Like Property} \label{par:Delta}
Let $\glos{\Delta}$ denote both the ``braid cloning''
operation $\Delta\colon \wB_n\to\wB_n\times\wB_n$ defined by
$B\mapsto (B,B)$ and the ``co-product'' algebra morphism \linebreak
$\Delta\colon \calA^w_n\to\calA^w_n\otimes\calA^w_n$ defined by cloning
permutations (that is, $\tau\mapsto\tau\otimes\tau$) and by treating
arrows as primitives (that is, \linebreak $a_{ij}\mapsto a_{ij}\otimes 1+1\otimes
a_{ij}$). Then the diagram on the right commutes. In formulae, this is
$\Delta(Z(B))=Z(B)\otimes Z(B)$, which is the statement ``$Z(B)$ is
group-like''.
\parpic[r]{$\xymatrix{
\wB_n \ar[r]^<>(0.5)\iota \ar[d]_Z & \wB_{n+1} \ar[d]^Z \\
\calA^w_n \ar[r]_<>(0.5)\iota & \calA^w_{n+1}
\ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Strand Insertions} \label{par:iota}
Let $\iota\colon \wB_n\to\wB_{n+1}$ be an operation of ``inert strand
insertion''. Given $B\in\wB_n$, the resulting $\iota B\in\wB_{n+1}$
will be $B$ with one strand $S$ added at some location chosen in
advance --- to the left of all existing strands, or to the right, or
starting from between the 3rd and the 4th strand of $B$ and ending
between the 6th and the 7th strand of $B$; when adding $S$, add it
``inert'', so that all crossings on it are virtual (this is well
defined). There is a corresponding inert strand addition operation
$\iota\colon \calA^w_n\to\calA^w_{n+1}$, obtained by adding a strand at the
same location as for the original $\iota$ and adding no arrows. It is
easy to check that $Z$ is compatible with $\iota$; namely, that the
diagram on the right is commutative.
\parpic[r]{$\xymatrix{
\wB_n \ar[r]^<>(0.5){d_k} \ar[d]_Z & \wB_{n-1} \ar[d]^Z \\
\calA^w_n \ar[r]_<>(0.5){d_k} & \calA^w_{n-1}
\ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Strand Deletions} \label{par:deletions} Given $1 \leq k \leq n$,
let $\glos{d_k}\colon \wB_n\to\wB_{n-1}$ be the operation of
``removing the $k$th strand''. This operation induces a homonymous
operation $d_k\colon \calA^w_n\to\calA^w_{n-1}$: if $D\in\calA^w_n$ is an
arrow diagram, then $d_kD$ is $D$ with its $k$th strand removed if no arrows
in $D$ start or end on the $k$th strand, and it is $0$ otherwise. It
is easy to check that $Z$ is compatible with $d_k$; namely, that the
diagram on the right is commutative.\footnote{In \cite{Bar-NatanDancso:WKO2} we'll
say that ``$d_k\colon \wB_n\to\wB_{n-1}$''
is an algebraic structure made of two spaces ($\wB_n$ and $\wB_{n-1}$),
two binary operations (braid composition in $\wB_n$ and in $\wB_{n-1}$),
and one unary operation, $d_k$. After projectivization we get the
algebraic structure $d_k\colon \calA^w_n\to\calA^w_{n-1}$ with $d_k$
as described above, and an alternative way of stating our assertion is
to say that $Z$ is a morphism of algebraic structures. A similar remark
applies (sometimes in the negative form) to the other operations discussed
in this section.}
\parpic[r]{$\xymatrix{
F_n \ar@{}[r]|{\mathsize{\Huge}{\actsonright}} \ar[d]_Z & \wB_n \ar[d]^Z \\
\FA_n \ar@{}[r]|{\mathsize{\Huge}{\actsonright}} & \calA^w_n
\ar@{}[ul]|{\text{\huge$\circlearrowleft$}}
}$}
\paragraph{Compatibility with the Action on $F_n$} \label{par:action}
Let $\glos{\FA_n}$ denote the (degree-completed) free, associative (but
not commutative) algebra on the generators $\glos{x_1,\dots,x_n}$. Then
there is an ``expansion'' $Z\colon F_n\to \FA_n$ defined by $\xi_i\mapsto
e^{x_i}$ (see~\cite{Lin:Expansions} and the related ``Magnus Expansion''
of~\cite{MagnusKarrasSolitar:CGT}). Also, there is a right action\footnote{In the language of
\cite{Bar-NatanDancso:WKO2}, we will say that $\FA_n=\proj F_n$ and
that when the actions involved are regarded as instances of the algebraic
structure ``one monoid acting on another'', we have that \linebreak
$\left(\FA_n\actsonright\calA^w_n\right)=\proj\left(F_n\actsonright
\wB_n\right)$.} of
$\calA^w_n$ on $\FA_n$ defined on generators by $x_i\tau=x_{\tau i}$
for $\tau\in S_n$ and by $x_ja_{ij}=[x_i,x_j]$ and $x_ka_{ij}=0$ for
$k\neq j$ and extended by the Leibniz rule to the rest of $\FA_n$ and
then multiplicatively to the rest of $\calA^w_n$.
\begin{exercise} Use the definition of the action in
\eqref{eq:ConceptualPsi} and the commutative diagrams of paragraphs
\ref{par:theta}, \ref{par:Delta} and~\ref{par:iota} to show that the
diagram of paragraph~\ref{par:action} is also commutative.
\end{exercise}
\pagebreak[2]
\parpic[r]{$\begin{array}{c}
\pstex{StrandDoubling} \\
\xymatrix{
\wB_n \ar[r]^<>(0.5){u_k} \ar[d]_Z & \wB_{n+1} \ar[d]^Z \\
\calA^w_n \ar[r]_<>(0.5){u_k} & \calA^w_{n+1}
\ar@{}[ul]|{\text{\huge$\not\circlearrowleft$}}
}
\end{array}$}
\paragraph{Unzipping a Strand} \label{par:unzip} Given $k$ between $1$ and
$n$, let $\glos{u_k}\colon \wB_n\to\wB_{n+1}$ the operation of ``unzipping
the $k$th strand'', briefly defined on the right\footnote{Unzipping
a knotted zipper turns a single band into two parallel ones. This
operation is also known as ``strand doubling'', but for compatibility with
operations that will be introduced later, we prefer ``unzipping''.}. The
induced operation $u_k\colon \calA^w_n\to\calA^w_{n+1}$ is also shown on
the right --- if an arrow starts (or ends) on the strand being doubled,
it is replaced by a sum of two arrows that start (or end) on either
of the two ``daughter strands'' (and this is performed for each arrow
independently; so if there are $t$ arrows touching the $k$th strands in
a diagram $D$, then $u_kD$ will be a sum of $2^t$ diagrams).
In some sense, much of this current series of papers as well as
the works of Kashiwara and Vergne~\cite{KashiwaraVergne:Conjecture}
and Alekseev and Torossian~\cite{AlekseevTorossian:KashiwaraVergne}
are about coming to grips with the fact that $Z$ is {\bf not}
compatible with $u_k$ (that the diagram on the right is {\bf not}
commutative). Indeed, let $x:=a_{13}$ and $y:=a_{23}$ be as on the
right, and let $s$ be the permutation $21$ and $\tau$ the permutation
$231$. Then $d_1Z(\overcrossing)=d_1(e^{a_{12}}s)=e^{x+y}\tau$
while $Z(d_1\overcrossing)=e^ye^x\tau$. So the failure of $d_1$
and $Z$ to commute is the ill-behaviour of the exponential function
when its arguments are not commuting, which is measured by the
BCH formula, central to both~\cite{KashiwaraVergne:Conjecture}
and~\cite{AlekseevTorossian:KashiwaraVergne}.
\subsubsection{Power and Injectivity} The following theorem is due to
Berceanu and
Papadima~\cite[Theorem~5.4]{BerceanuPapadima:BraidPermutation}; a variant of
this theorem are also true for ordinary braids~\cite{Kohno:deRham, Bar-Natan:Homotopy,
HabeggerMasbaum:Milnor}, and can be proven by similar means:
\begin{theorem} $Z\colon \wB_n\to\calA^w_n$ is injective. In other words, finite
type invariants separate w-braids.
\end{theorem}
\begin{proof}The statement follows immediately from the faithfulness of
the action $F_n\actsonright\wB_n$, from the compatibility
of $Z$ with this action, and from the injectivity
of $Z\colon F_n\to\FA_n$ (the latter is well known, see
e.g.~\cite[Theorem~5.6]{MagnusKarrasSolitar:CGT}\footnote{Though notice
that we use $\xi_i\mapsto e^{x_i}$ whereas
\cite[Theorem~5.6]{MagnusKarrasSolitar:CGT} uses $\xi_i\mapsto 1+x_i$. The
\cite{MagnusKarrasSolitar:CGT} injectivity proof holds in our case
just as well.} and~\cite{Lin:Expansions}). Indeed, if $B_1$ and $B_2$
are w-braids and $Z(B_1)=Z(B_2)$, then $Z(\xi)Z(B_1)=Z(\xi)Z(B_2)$ for
any $\xi\in F_n$. Therefore $\forall\xi\, Z(\xi\sslash B_1)=Z(\xi\sslash
B_2)$, therefore $\forall\xi\,\xi\sslash B_1=\xi\sslash B_2$, therefore
$B_1=B_2$. \qed
\end{proof}
\begin{remark} Apart from the easy fact that $\calA^w_n$ can be computed
degree by degree in exponential time, we do not know a simple formula for
the dimension of the degree $m$ piece of $\calA^w_n$ or a natural basis of
that space. This compares unfavourably with the situation for ordinary
braids (see e.g.~\cite{Bar-Natan:Braids}). Also compare with
Problem~\ref{prob:wCombing} and with Remark~\ref{rem:GutierrezKrstic}.
\end{remark}
\subsubsection{Uniqueness} There is certainly not a unique expansion for
w-braids --- if $Z_1$ is an expansion and and $P$ is any degree-increasing
linear map $\calA^w\to\calA^w$ (a ``pollution'' map), then $Z_2:=(I+P)\circ
Z_1$ is also an expansion, where $I\colon \calA^w\to\calA^w$ is the
identity. But that's all, and if we require a bit more, even that
freedom disappears.
\begin{proposition} If $Z_{1,2}\colon \wB_n\to\calA^w_n$ are expansions then
there exists a degree-increasing linear map $P\colon \calA^w\to\calA^w$ so
that $Z_2=(I+P)\circ Z_1$.
\end{proposition}
\begin{proof} (Sketch). Let $\widehat{\wB_n}$ be the unipotent completion
of $\wB_n$. That is, let $\bbQ\wB_n$ be the algebra of formal linear
combinations of w-braids, let $\calI$ be the ideal in $\bbQ\wB_n$ be the
ideal generated by $\semivirtualover=\overcrossing-\virtualcrossing$ and by
$\semivirtualunder=\virtualcrossing-\undercrossing$, and set
\[ \widehat{\wB_n}:=
\underleftarrow{\lim}_{m\to\infty} \bbQ\wB_n \left/\calI^m\right..
\]
$\widehat{\wB_n}$ is filtered with
$\calF_m\widehat{\wB_n}:=\underleftarrow{\lim}_{m'>m} \calI^m
\left/\calI^{m'}\right..$ An ``expansion'' can be re-interpreted as an
``isomorphism of $\widehat{\wB_n}$ and $\calA^w_n$ as filtered vector
spaces''. Always, any two isomorphisms differ by an automorphism of the
target space, and that's the essence of $I+P$. \qed
\end{proof}
\begin{proposition} If $Z_{1,2}\colon \wB_n\to\calA^w_n$ are homomorphic
expansions that commute with braid cloning (Paragraph~\ref{par:Delta}) and
with strand insertion (Paragraph~\ref{par:iota}), then \linebreak $Z_1=Z_2$.
\end{proposition}
\begin{proof} (Sketch). A homomorphic expansion that commutes with strand
insertions is determined by its values on the generators $\overcrossing$,
$\undercrossing$ and $\virtualcrossing$ of $\wB_2$. Commutativity
with braid cloning (see Paragraph \ref{par:Delta}) implies that these values must be, up to permuting
the strands, group like: that is, the exponentials of primitives. But
the only primitives are $a_{12}$ and $a_{21}$, and one may easily
verify that there is only one way to arrange these so that $Z$
will respect $\virtualcrossing^2=\overcrossing\cdot\undercrossing=1$ and
$\semivirtualover\mapsto\rightarrowdiagram+(\text{higher degree terms})$. \qed
\end{proof}
\subsubsection{The Group of Non-Horizontal Flying Rings}
\label{subsubsec:NonHorRings}
Let
$\glos{Y_n}$ denote the space of all placements of $n$ numbered disjoint
oriented unlinked geometric circles in $\bbR^3$. Such a placement
is determined by the centres in $\bbR^3$ of the circles, the radii,
and a unit normal vector for each circle pointing in the positive
direction, so $\dim Y_n=3n+n+3n=7n$. $S_n \ltimes \bbZ_2^n$ acts on
$Y_n$ by permuting the circles and mapping each circle to its image in
either an orientation-preserving or an orientation-reversing way. Let
$\glos{\tilde{Y}_n}$ denote the quotient $Y_n/S_n \ltimes \bbZ_2^n$.
The fundamental group $\pi_1(\tilde{Y}_n)$ can be thought of as the
``group of flippable flying rings''. Without loss of generality, we
can assume that the basepoint is chosen to be a horizontal placement.
We want to study the relationship of this group to $\wB_n$.
\begin{theorem}
$\pi_1(\tilde{Y}_n)$ is a $\bbZ_2^n$-extension of $\wB_n$, generated
by $s_i$, $\sigma_{i}$ ($1\leq i \leq n-1)$, and $\glos{w_i}$ (``flips''),
for $1\leq i \leq n$; with the relations as above, and in addition:
\[
w_i^2=1; \qquad w_iw_j=w_jw_i; \qquad w_js_i=s_iw_j \quad \text{when } \quad i\neq j, j+1;
\]
\[
w_is_i=s_iw_{i+1}; \qquad w_{i+1}s_i=s_iw_i; \quad w_j\sigma_{i}=\sigma_{i}w_j \quad \text{if } \quad j \neq i, i+1;
\]
\[
w_{i+1}\sigma_{i}=\sigma_{i}w_{i}; \quad \text{yet } \quad
w_i\sigma_{i}=s_i\sigma_i^{-1}s_iw_{i+1}.
\]
\end{theorem}
The two most interesting flip relations in pictures:
\begin{equation}\label{eq:FlipRels}
\raisebox{-10mm}{\input figs/FlipRels.pstex_t}
\end{equation}
\parpic[r]{\input{figs/FlippingRing.pstex_t}}
Instead of a proof, we provide some heuristics.
Since each circle starts out in a horizontal position and returns
to a horizontal position, there is an integer number of
``flips'' they do in between, these are the generators $w_i$, as
shown on the right.
The first relation says that a double flip is homotopic to doing nothing.
Technically, there are two different directions of flips, and they are the
same via this (non-obvious but true) relation. The rest of the first line is
obvious: flips of different rings commute, and if
two rings fly around each other while another one flips, the order of these
events can be switched by homotopy. The second line says that, if two rings trade
places with no interaction while one flips, then the order of these events can be
switched as well. However, we have to re-number the flip to conform to the
strand labelling convention.
The only subtle point is how flips interact with crossings. First of all,
if one ring flies through another while a third one flips, the order clearly
does not matter. If a ring flies through another and also flips, the
order can be switched. However, if ring $A$ flips and then ring $B$ flies
through it, this is homotopic to ring $B$ flying through ring $A$
from the other direction and then ring $A$ flipping. In other words, commuting
$\sigma_i$ with $w_i$ changes the ``sign of the crossing'' in the sense of
Exercise \ref{ex:swBn}. This gives the last, and the only truly non-commutative flip
relation.
\parpic[r]{\input{figs/Wen.pstex_t}}
To explain why the flip is denoted by $w$, let us consider the alternative
description by ribbon tubes. A flipping ring traces a so called
wen\footnote{The term wen was coined by Kanenobu and Shima in
\cite{KanenobuShima:TwoFiltrationsR2K}}
in $\bbR^4$. A wen is a Klein bottle cut along a meridian circle,
as shown. The wen is embedded in $\bbR^4$.
Finally, note that $\pi_1Y_n$ is exactly the pure $w$-braid group
$\PwB_n$: since each ring has to return to its original position and
orientation, each does an even number of flips. The flips (or wens)
can all be moved to the bottoms of the braid diagram strands (to the
bottoms of the tubes, to the beginning of words), at a possible cost, as
specified by Equation~\eqref{eq:FlipRels}. Once together, they pairwise
cancel each other. As a result, this group can be thought of as not
containing wens at all.
\subsubsection{The Relationship with u-Braids} \label{subsubsec:RelWithu}
For
the sake of ignoring strand permutations, we restrict our
attention to pure braids.
\parpic[r]{$\xymatrix{
\PuB \ar@{.>}[r]^{Z^u} \ar[d]^a & \calA^u \ar[d]^\alpha \\
\PwB \ar[r]^{Z^w} & \calA^w
}$}
By Section \ref{subsubsec:FTAlgebraic}, for any expansion $Z^u\colon
\PuB_n \to \calA^u_n$ (where $\PuB_n$ is the ``usual'' braid group
and $\calA^u_n$ is the algebra of horizontal chord diagrams on $n$
strands), there is a square of maps as shown on the right. Here $Z^w$
is the expansion constructed in Section~\ref{subsec:wBraidExpansion},
the left vertical map $\glos{a}$ is the composition of the inclusion
and projection maps $\PuB_n \to \PvB_n \to \PwB_n$. The map $\glos{\alpha}$
is the induced map by the functoriality of projectivization, as noted
after Exercise \ref{ex:BraidsAlgApproach}. The reader can verify that
$\alpha$ maps each chord to the sum of its two possible directed versions.
Note that this square is {\it not} commutative for any choice of $Z^u$ even
in degree 2: the image of a crossing under $Z^w$ is outside the image
of $\alpha$.
\parpic[r]{\input{figs/uwsquare2.pstex_t}}
More specifically, for any choice $c$ of a ``parenthesization'' of $n$
points, the KZ-construction / Kontsevich integral (see for example
\cite{Bar-Natan:NAT}) returns an expansion $Z_c^u$ of $u$-braids. We shall
see in \cite{Bar-NatanDancso:WKO2} (Proposition 4.15 there)
that for any choice of $c$, the two
compositions $\alpha \circ Z_c^u$ and $Z^w \circ a$ are ``conjugate in a
bigger space'': there is a map $i$ from $\calA^w$ to a larger space of
``non-horizontal arrow diagrams'', and in this space the images of the
above composites are conjugate. However, we are not certain that $i$
is an injection, and whether the conjugation leaves the $i$-image of
$\calA^w$ invariant, and so we do not know if the two compositions differ
merely by an outer automorphism of $\calA^w$.
| 42,856
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Secure your investment by setting in place the three keys to strategic implementation
Once you have a strategic plan in place, a powerful tool to monitor your progress and communicate company direction is performance management software, like OnStrategy’s implementation module. Before you choose a system for managing performance, it is important to make sure that you’ll get the most out of it. In order to make the most of your software, you’ll need to avoid some common strategy pitfalls by putting in place some time-honored business practices.
The hard truth: without proper management, your performance management software could become just another program on your desktop
Many businesses, even those with well-made plans, fail to implement their strategy. Their problem lies in ineffectively managing their employees once their plan is in place. Sure, they’ve conducted surveys, collected data, gone on management retreats to decide on their organization’s direction- even purchased expensive software to manage their process- but somewhere their plan falls apart.
Expensive performance management software does not ensure success: Implementation of your plan depends on solid accountability
Accountability is key to successful implementation of your business strategy— hands down..
Before you purchase software to manage your organization’s performance, make sure you set in place these 3 keys to implementation:
With the right manager, the right rewards and the right coaching, performance management software can be the most powerful tool for making sure your plan becomes a reality.
>> Continue reading with Step One: Appointing a Strategic Plan Manager
| 331,339
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Simply put…real talk…this new one from DJ Soko, “Stand Up” which features production from the always steady Apollo Brown and furious lyricism from one of the most slept-on emcees EVER, Hassaan Mackey and Guilty Simpson, is undoubtedly my favorite track to drop in 2012. This jawn stayed on repeat for 23(!) consecutive listens while at the gym yesterday. Isn’t it amazing what a simple M.O.P. vocal slice can do to “amp up” any track!?
DJ Soko and Apollo Brown, two thirds of acclaimed group The Left, command you to “Stand Up.” For this brazen order, Soko enlists Brown on the boards and the incomparable mic skills of Guilty Simpson and Hassaan Mackey.
The 4th installment in Mello Music Group’s monthly 7” series finds Apollo finessing a triumphant horn loop that would make Pete Rock proud, while Soko adds personality to the beat via a scratch chorus made up of declaratives from past classics by Pharoahe Monch and M.O.P.. Motor City rhyme animal Guilty Simpson stomps into the track, providing a sixteen bar reminder why he’s one of the fiercest on the mic today. Low Budget’s Hassaan Mackey takes the baton from Guilt, inviting you to take his lines “like a razor to the throat.” Two raw MC’s going for theirs over an equally unprocessed beat; there is little gloss present here, this is meat-and-potatoes rap for hungry heads.
Another brick in the foundation of the growing Mello Music Group legacy, “Stand Up” showcases the expert handle on the fundamentals of hip-hop that the MMG squad holds.
| 220,619
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Corporate Gifting
At Vedantaa Spa & Wellness we understand that it’s important to thank those who have supported you and allowed you to continue with your business. We help foster goodwill with your clients and accounts.
Now is the time to thank all those in your world who have made a meaningful contribution to your bottom line and be seen as a sincere and generous company by gifting in the right way.
Call us on Toll Free Number:- 1-800-3000-3141 for your Corporate Gifting enquiries.
| 24,626
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I am trying to comment on this blog:
But my captcha entry is not being accepted and I have entered it correctly countless times. I keep getting the following message:
Error: You entered in the wrong CAPTCHA phrase. Press your browser's back button and try again.
Why won't WordPress accept my correct CAPTCHA entry?
The blog I need help with is arunareject.wordpress.com.
| 205,501
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I’ve been trying to work with a group of fellow alumni on a game that will be our “break-in” indie game now that we’re out of school. We are following all the best advice on start-up studios, researching the business end of things as well as the development end, and laying the foundations of good communication.
You’ll notice I said “trying” to work. We hit a bit of a snag in the start that we shouldn’t have even run into. What was it? We tried to make a game that was logical for us to make, rather than something we WANTED to make. We opted for the safe bet rather than the dream. We should have known better… after all we got our degrees so we could make games we loved.
So what changed?
I was speaking with one of the lovely professors that so recently released us into the wild and he said we should come together as a team into a game we could all get behind. He said we’d crash and burn if we didn’t love what we were working on. (I’m paraphrasing of course.)
So I decided that we needed to find out what we all loved. There are six of us, and we’re working remotely, so finding common ground isn’t easy. I decided to go with a survey:
Looks simple, right? Sometimes it’s the simple things that trip us up, like trying to make a game that we didn’t care much about.
Turns out we ALL wanted to make RPGs (a few had multiple genres, but RPG was on everybody’s list.) We had been trying to make a RTS game. We also all loved the same general themes and periods. We trashed the RTS game completely and started over.
Now we are rolling. The over-arching story is fitting into place, the mechanics are being hashed out, and the technical details of platform and engine are being looked at. This is looking like it’ll be a great experience after all, and if all goes well a brand new indie studio will be rearing its head within the next year.
Watch for us. 😉
| 292,425
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CSEC watchdog muzzled, defanged: Greg Weston
The wish and 'a prayer' of keeping tabs on CSEC
The revelation that a little-known Canadian intelligence operation has been electronically spying on trading partners and other nations around the world, at the request of the U.S. National Security Agency, has critics wondering who's keeping an eye on our spies..
The Harper government recently appointed a new oversight commissioner for Canada's electronic spy agency, the Communications Security Establishment Canada. But he will be only part-time until next April.
Even then, Senator Hugh Segal, the chief of staff to former Conservative prime minister Brian Mulroney and someone with a long involvement in security intelligence issues, says any notion of effective public oversight of Canada's electronic spying agency is "more like a prayer" than fact.
The debate over who's keeping tabs on our spies has heightened in recent days following a CBC News report detailing a top secret document retrieved by American whistleblower Edward Snowden.
The document shows that the agency known as CSEC set up covert spying posts around the world at the request of the giant NSA.
Both agencies gather intelligence by intercepting mostly foreign phone calls and hacking into computer systems around the world.
U.S. President Barack Obama has ordered a widespread investigation of the NSA after leaked Snowden documents revealed the agency was gathering massive amounts of information on millions of American citizens.
In this country, the Harper government simply keeps pointing to CSEC's oversight commissioner as proof that Canadians have nothing to worry about.
As Defence Minister Rob Nicholson told the Commons this week: "There is a commissioner that looks into CSEC [and] every year for 16 years has confirmed that they've acted within lawful activities."
Well, not exactly.
'Contrary to law'
Only months ago, the recently retired CSEC commissioner, Justice Robert Decary, stated in his final report that he had uncovered records suggesting some of CSEC's spying activities "may have been directed at Canadians, contrary to law."
The retired justice said the CSEC records were so unclear or incomplete that he was unable to determine whether the agency had been operating legally.
Decary's predecessor, Justice Charles Gonthier, filed the same complaint about incomplete or missing records in his day, which forced him to report in a similar fashion that he could not determine if CSEC had been breaking the law.
Gonthier also alluded to a CSEC operation in 2006 that he suggested may have been illegal.
The head of CSEC at the time, John Adams, recently told CBC News that, as a result of that discovery, "I shut the place down for a while."
However, intelligence experts have told CBC News that the oversight problems at CSEC are much deeper than poor record-keeping.
They say successive commissioners have simply lacked both the resources and the legal mandate to conduct meaningful oversight.
The current commissioner, Judge Jean-Pierre Plouffe, operates with a staff of 11, about half of whom actually work on investigations, largely to ensure CSEC isn't abusing its powers by spying on Canadians.
But CSEC employs over 2,000 people who covertly collect masses of information recently described as more data per day than all the country's banking transactions combined.
As Segal says, the result is obvious: "When there are thousands of people at CSEC processing millions of messages every day of all kinds, the notion that a group of 11 might be able to provide proper oversight is more like a prayer than any kind of constructive statement of fact."
Not exactly as written
Of course, even if a commissioner did discover something seriously amiss at the electronic eavesdropping agency, there is a chance Canadians would never know.
Here's how the system works:
Suppose the commissioner's oversight sleuths discover that CSEC is illegally intercepting phone calls and hacking into the computers of certain Canadians.
The oversight commissioner is required to report his discovery in a top secret report to the defence minister.
That happens to be the same minister responsible for CSEC, and from whom the agency gets its government direction.
It is also the minister who would be at the centre of any CSEC scandal if news of this breach leaked out.
If the minister refuses to expose his own agency's wrongdoing, the oversight commissioner can try to use his annual report to Parliament to do that.
But a funny thing happens on the way to Parliament.
First, CSEC gets to censor the entire report. Then it goes back to the same defence minister.
The minister is required to present the sanitized version of the report to Parliament, but has no obligation to mention it is not exactly as originally written.
Former CSEC chief Adams admits the agency is "very, very biased towards the less the public knows the better."
He points out that in the spying business, opening an agency's operations to full public scrutiny "would be kind of like unilateral disarmament, because if Canadians know everything CSEC can and can't do, then everyone else will too."
But as the leaked Snowden documents continue to force back the curtains at CSEC, Adams says it is time to find a better way to reassure Canadians about what they are doing.
"I think a knowledgeable Canadian is going to be much easier to deal with," he says.
If the public reaction to the Snowden revelations is any indication, Canadians are all.
| 408,814
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$350.00
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1,1':4',1'':4'',1''':4''',1'''':4'''',1'''''-Sexiphenyl
- Formula: C36H26
- Molecular weight: 458.5916
-
- IUPAC Standard InChIKey: ZEMDSNVUUOCIED-UHFFFAOYSA-N
- CAS Registry Number: 4499-83-6
- Chemical structure:
This structure is also available as a 2d Mol file or as a computed 3d SD file
The 3d structure may be viewed using Java or Javascript.
- Other names: p-Quaterphenyl, 4,4'''-diphenyl-; p-Sexiphenyl; 1,1':4',1'':4'',1'''-Quaterphenyl, 4,4'''-diphenyl-; p-Hexaphenyl; p-Sexiphen.
| 143,734
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TITLE: Probability that a random variable is among the top k out of n when ordered
QUESTION [1 upvotes]: Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$.
What is the probability distribution that any given $X_i$ is among the top $k$ largest of the $n$ $X$'s?
For example, there are $10$ individuals. Each of them draw a random number from a distribution $F(x)$ with density function $f(x)$. The draws are i.i.d. What is the probability that individual $i$'s draw will be among the top $3$ largest numbers (i.e. either top, second or third)?
Many thanks.
REPLY [0 votes]: $X_1,\ldots,X_n$ are the $n$ i.i.d. random variables.
Let $X_{(1)}<\cdots< X_{(n)}$ be the order statistics, i.e. the same random variables sorted. (By continuity of the c.d.f., we need not write "$\le$".)
Then $X_i=X_{(j)}$. Given $i$, what is the distribution of $j\text{ ?}$
Let $Y_1,\ldots,Y_n$ be $X_{\sigma(1)},\ldots,X_{\sigma(n)}$, where $\sigma$ is some permutation.
Lemma: The joint distribution of $Y_1,\ldots,Y_n$ is the same as the joint distribution of $X_1,\ldots,X_n$.
Hence $\Pr(Y_i = Y_{(j)})=\Pr(X_i=X_{(j)})$. But $\Pr(Y_i=Y_{(j)})$ is the probability that $X_{\sigma(i)}$ is in the $j$th position when sorted. Thus the probability that $X_{\sigma(i)}$ is in the $j$th position is the same as the probability that $X_i$ is in the $j$th position.
This works regardless of which permutation $\sigma$ is, so it shows that every index has the same probability of being in the $j$th position. This is true for every value of $j$.
Hence the rank of $X_i$ in the sorting is uninformly distributed on the set $\{1,\ldots,n\}$. It probability of being in any particular subset is the size of that subset divided by $n$.
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\section{Intersection products on tropical linear spaces} \label{sec-tropicallinearspaces}
In this section we will give a proof that tropical linear spaces
$L^n_k$ admit an intersection product. Therefore we show at first
that the diagonal in the Cartesian product $L^n_k \times L^n_k$ of
such a linear space with itself is a sum of products of Cartier
divisors. Given two cycles $C$ and $D$ we can then intersect the
diagonal with $C \times D$ and define the product $C \cdot D$ to
be the projection thereof.
Throughout the section $e_1,\dots,e_n$ will always be the standard
basis vectors in $\RR^n$ and $e_0 := -e_1- \ldots -e_n$.
We begin the section with our basic definitions:
\begin{definition}[Tropical linear spaces] \label{def-Lnk}
For $I \subsetneq
\{0,1,\dots,n\}$ let $\sigma_I$ be the cone generated by the
vectors $e_i$, $i \in I$. We denote by $L^n_k$ the tropical
fan consisting of all cones $\sigma_I$ with $I \subsetneq
\{0,1,\dots,n\}$ and $|I| \leq k$, whose maximal cones
all have weight one (cf. \cite[example 3.9]{AR07}).
This fan $L^n_k$ is a representative of
the tropical linear space $\max \{ 0, x_1,\ldots,x_n \}^{n-k}
\cdot \RR^n$.
\end{definition}
\begin{definition}
Let $C \in Z_k(\RR^n)$ be a tropical cycle and let the map $i: \RR^n
\rightarrow \RR^n \times \RR^n$ be given by $x \mapsto (x,x)$.
Then the push-forward cycle $$ \triangle_C := i_{*}(C) \in
Z_k(\RR^n \times \RR^n)$$ is called the \emph{diagonal} of $C \times C$.
\end{definition}
In order to express the diagonal in $L^n_k \times L^n_k$ by means
of Cartier divisors we first have to refine $L^n_k \times L^n_k$
in such a way that the diagonal is a subfan of this refinement:
\begin{definition}
Let $F^n_k$ be the refinement of $L^n_k \times L^n_k$ that arises recursively from
$L^n_k \times L^n_k$ as follows:
Let $M := (L^n_k \times L^n_k)^{(2k)}$ be the set of maximal
cones in $L^n_k \times L^n_k$.
If a cone $\sigma \in M$ is generated by
$$\left( \begin{array}{c} -e_{i} \\ \hline 0 \end{array} \right)
, \left( \begin{array}{c} 0 \\ \hline -e_i \end{array} \right), v_3, \ldots, v_{2k}$$
for some $i$ and vectors
$$v_j \in \left\{ \left. \left( \begin{array}{c} -e_\mu \\ \hline -e_\mu \end{array} \right),
\left( \begin{array}{c} -e_\mu \\ \hline 0 \end{array} \right),
\left( \begin{array}{c} 0 \\ \hline -e_\mu \end{array} \right) \right| \mu =0,\ldots,n \right\}$$
then replace the cone $\sigma$ by the two cones spanned by
$$\left( \begin{array}{c} -e_{i} \\ \hline -e_{i} \end{array} \right),
\left( \begin{array}{c} -e_{i} \\ \hline 0 \end{array} \right)
,v_3,\ldots,v_{2k}$$
and
$$\left( \begin{array}{c} -e_{i} \\ \hline -e_{i} \end{array} \right),
\left( \begin{array}{c} 0 \\ \hline -e_{i} \end{array} \right)
,v_3,\ldots,v_{2k},$$
respectively. Repeat this process until there are no more
cones in $M$ that can be replaced. The fan $F^n_k$ is then the set of
all faces of all cones in $M$.
\end{definition}
The next lemma provides a technical tool needed in the proofs of
the subsequent theorems:
\begin{lemma} \label{lem-zerofunctionzeroweight}
Let $F$ be a complete and smooth fan in $\RR^n$ (in the sense of toric geometry)
and let the weight of every maximal cone in $F$ be one. Moreover, let $h_1,\ldots,h_r$, $r \leq n$, be
rational functions on $\RR^n$ that are linear on every cone of
$F$. Then the intersection product $h_1 \cdots h_r \cdot F$ is
given by
$$h_1 \cdots h_r \cdot F = \left( \bigcup_{i=0}^{n-r} F^{(i)},\omega_{h_1 \cdots h_r} \right)$$
with some weight function $\omega_{h_1 \cdots h_r}$ on the cones
of dimension $n-r$.\\
Let $\tau \in F^{(n-r)}$ be a cone in $F$ such that for all
maximal cones $\sigma \in F^{(n)}$ with $\tau \subseteq \sigma$ there exists
some index $i \in \{1,\ldots,r \}$ such that $h_i$ is identically zero on $\sigma$. Then
holds: $$\omega_{h_1 \cdots h_r}(\tau)=0.$$
\end{lemma}
\begin{proof}
We proof the claim by induction on $r$: For $r=1$ we are in the
situation that $h_1$ is identically zero on every maximal cone adjacent to
$\tau$. Hence $\omega_{h_1}(\tau)=0$. Now let $r>1$. Using the
induction hypothesis we can conclude that $|h_1 \cdots h_{r-1} \cdot
F| \subseteq \bigcup_{\sigma \in S} \sigma$, where $$S:= {\{\sigma \in F^{(n)} | \text{none of } h_1, \ldots, h_{r-1} \text{ is identically zero on
} \sigma \}}.$$ Our above assumption then implies that $h_r$ must be identically
zero on every cone in $$\{ \sigma \in F^{(n)} | \tau \subseteq \sigma \text{ and none of } h_1, \ldots, h_{r-1} \text{ is identically zero on } \sigma
\}$$ and thus that $\omega_{h_1 \cdots h_r}(\tau)=0$.
\end{proof}
\begin{notation} \label{notation-vector_as_rat_function}
Let $F$ be a simplicial fan in $\RR^n$ and let $u$ be
a generator of a ray $r_u$ in $F$. By abuse of notation we also denote by
$u$ the unique rational function on $|F|$ that is linear on
every cone in $F$, that has the value one on $u$ and that is
identically zero on all rays of $F$ other than $r_u$.
If not stated otherwise, vectors considered as Cartier divisors will from
now on always denote rational functions on the complete fan $F^n_n$.
\end{notation}
\begin{notation}
Let $C$ be a tropical cycle and let $h_1,\ldots,h_r \in \Div(C)$
be Cartier divisors on $C$. If $$P(x_1,\ldots,x_r)=\sum_{i_1+\ldots+i_r \leq d}
\alpha_{i_1,\ldots,i_r} x_1^{i_1} \cdots x_r^{i_r}$$ is a polynomial
in variables $x_1,\ldots,x_r$ we denote by $P(h_1,\ldots,h_r) \cdot C$
the intersection product
$$P(h_1,\ldots,h_r) \cdot C := \sum_{i_1+\ldots+i_r \leq d}
\left( \alpha_{i_1,\ldots,i_r} h_1^{i_1} \cdots h_r^{i_r} \cdot C \right).$$
\end{notation}
In the following theorem we give a description of the diagonal
$\triangle_{L^n_{n-k}}$ by means of Cartier divisors on our fan
$F^n_n$:
\begin{theorem}\label{thm-diagonalinRn}
The fan
$$ \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\dots \left( \left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right) \cdot
\left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\cdot F^n_n $$
is a representative of the diagonal $\triangle_{L^n_{n-k}}$.
\end{theorem}
\begin{proof}
First of all, note that $$\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) $$ is
a representation of the tropical polynomial $\max \{0,x_1,\ldots,x_n \}$, where $x_1,\ldots,x_n$
are the coordinates of the first factor of $\RR^n \times \RR^n$.
Applying \cite[lemma 9.6]{AR07} we obtain
$$\left[ \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k \cdot F^n_n \right]= [L^n_{n-k} \times \RR^n].$$
By lemma \cite[lemma 9.4]{AR07} we can conclude that
$\triangle_{\RR^n} \cdot [L^n_{n-k} \times \RR^n] =
i_{*}([L^n_{n-k}])=\triangle_{L^n_{n-k}}$ and hence it suffices to show
that $[X]=\triangle_{\RR^n}$ for
$$ X:= \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\dots \left( \left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)
\right) \cdot F^n_n$$
to prove the claim. Therefore, let $\sigma = \langle r_1, \ldots, r_n \rangle_{\RR_{\geq 0}} \in X^{(n)}$
be a cone not contained in $|\triangle_{\RR^n}|$. We will show that the weight of $\sigma$ in $X$ has to be zero.
W.l.o.g. we assume that $$r_1 \not\in
D:=\left\{ \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right),\dots,
\left( \begin{array}{c} -e_n \\ \hline -e_n \end{array}
\right) \right\}.$$
Moreover, let $$T:= \left\{ \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right),\dots,
\left( \begin{array}{c} -e_n \\ \hline 0 \end{array}
\right) \right\} \text{ and } B:=\left\{ \left( \begin{array}{c} 0 \\ \hline -e_1 \end{array} \right),\dots,
\left( \begin{array}{c} 0 \\ \hline -e_n \end{array}
\right) \right\}.$$
We distinguish between two cases:
\begin{enumerate}
\item[1.] First, we assume that $$r_i \not\in \left\{\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right),
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)
\right\}, i=1,\ldots,n. $$
Changing the given rational functions by globally linear functions we can rewrite the above intersection product as $X=\varphi_1
\cdots \varphi_n \cdot F^n_n$, where
$$\varphi_i = \left\{ \begin{array}{ll}
\left( \begin{array}{c} -e_i \\ \hline 0 \end{array} \right)+\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right), & \text{if } \left( \begin{array}{c} -e_i \\ \hline 0 \end{array} \right) \not\in \{r_1,\ldots,r_n\} \vspace{2mm} \\
\left( \begin{array}{c} 0 \\ \hline -e_i \end{array} \right)+\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right), & \text{else.}
\end{array} \right.$$
Now we apply lemma \ref{lem-zerofunctionzeroweight}: If the weight of $\sigma$ in $X$
is non-zero there must be at least one cone $$\widetilde{\sigma} =\langle r_1,\ldots,r_n,v_1,\dots,v_n
\rangle_{\RR \geq 0} \in F^n_n$$ such that all rational functions $\varphi_1,\ldots,\varphi_n$
are non-zero on $\widetilde{\sigma}$. We study three subcases:
\begin{enumerate}
\item There are vectors $r_i \in
T$ and $r_j \in B$: Then we need both vectors $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)$ and $\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)$ among the $v_\mu$ such that all functions $\varphi_i$
are non-zero on $\widetilde{\sigma}$. But there is no cone in
$F^n_n$ containing these two vectors.
\item $r_1 \in T$ (or $r_1 \in B$) and $r_j \in D$ for some $j$ and $r_i \in T \cup D$ (or $r_i \in B \cup D$) for all $i$:
As there is no cone in $F$ containing $\left( \begin{array}{c} -e_i \\ \hline 0 \end{array}
\right)$ and $\left( \begin{array}{c} 0 \\ \hline -e_i \end{array}
\right)$ for any $i$, we need $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)$ among the $v_\mu$ such that all functions $\varphi_i$ are non-zero on $\widetilde{\sigma}$.
Moreover, if $\left( \begin{array}{c} -e_i \\ \hline 0 \end{array} \right) \not\in \{r_1,\ldots,r_n\}$ then
we must have $\left( \begin{array}{c} -e_i \\ \hline 0 \end{array}
\right) \in \{v_1,\ldots,v_n\}$. But there is no cone in $F^n_n$
containing $\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array}
\right), \ldots, \left( \begin{array}{c} -e_n \\ \hline 0 \end{array}
\right)$ and $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)$. (Analogously for $B$, but with $\varphi_i$ defined the other way around.)
\item All vectors $r_i$ are contained in $T$ (or in $B$):
In this case we need $\left( \begin{array}{c} 0 \\ \hline -e_1 \end{array}
\right)$ or $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)$ among the $v_\mu$ such that all functions $\varphi_i$
are non-zero, but again there is no such cone.
(Analogously for $B$, but with $\varphi_i$ defined the other way around.)
\end{enumerate}
\item[2.] Now we assume that $$r_1= \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right) \text{ } \left( \text{or } r_1= \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right) \right).$$
Like before we rewrite the intersection product as $X=\varphi_1
\cdots \varphi_n \cdot F^n_n$ with $\varphi_i$ defined as
above and apply lemma \ref{lem-zerofunctionzeroweight}:
If $\left( \begin{array}{c} -e_i \\ \hline 0 \end{array} \right) \not\in
\{r_1,\ldots,r_n\}$ then $\varphi_i=\left( \begin{array}{c} -e_i \\ \hline 0 \end{array}
\right)+\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)$ and we need $\left( \begin{array}{c} -e_i \\ \hline 0 \end{array}
\right)$ or $\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)$ among the $v_\mu$ such that all functions $\varphi_i$ are non-zero on $\widetilde{\sigma}$.
But as there is no cone in $F^n_n$ containing
$\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)$
and $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)$ we must have $\left( \begin{array}{c} -e_i \\ \hline 0 \end{array}
\right) \in \{v_1,\ldots,v_n\}$. Hence all the vectors $\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array}
\right),\ldots,\left( \begin{array}{c} -e_n \\ \hline 0 \end{array}
\right)$ and $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)$ must be contained in
$\{r_1,\ldots,r_n,v_1,\ldots,v_n\}$, but there is no such cone
in $F^n_n$. (Analogously for $r_1= \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)$, but with $\varphi_i$ defined the other way around.)
\end{enumerate}
So far we have proven that our intersection cycle $X$ is
contained in the diagonal $\triangle_{\RR^n}$. As the diagonal is
irreducible we can then conclude by \cite[lemma 2.21]{GKM07} that
$[X]=\lambda \cdot \triangle_{\RR^n}$ for some integer $\lambda$.
Thus our last step in this proof is to show that $\lambda=1$:
Let $\varphi_1,\ldots,\varphi_n$ be the rational functions given
above. We obtain the following equation
of cycles in $\RR^n \times \RR^n$:
$$\begin{array}{rcl}
&& \varphi_1 \cdots \varphi_n \cdot [\{0\} \times \RR^n] \vspace{2mm}\\
&=& \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\dots \left( \left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)
\right) \cdot [\{0\} \times \RR^n] \vspace{2mm}\\
&=& \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^n \cdot [\{0\} \times \RR^n] \vspace{2mm}\\
&=& \{0\} \times \{0\}.
\end{array}$$
As $\varphi_1 \cdots \varphi_n \cdot [\RR^n \times
\RR^n]= \lambda \cdot \triangle_{\RR^n}$, by \cite[definition 9.3]{AR07} and \cite[remark 9.9]{AR07}
we obtain the equation
$$\begin{array}{rcl}
\lambda \cdot \{0\} &=& \lambda \cdot ( \{0\} \cdot \RR^n)\\
&=& \pi_{*}(\varphi_1 \cdots \varphi_n \cdot (\{0\} \times \RR^n) )\\
&=& \pi_{*}(\{0\} \times \{0\}) )\\
&=& 1 \cdot \{0\}
\end{array}$$
of cycles in $\RR^n$. This finishes the proof.
\end{proof}
Our next step is to derive a description of the diagonal
$\triangle_{L^n_{n-k}}$ on $L^n_{n-k} \times L^n_{n-k}$ from our
description on $F^n_n$:
\begin{theorem}\label{thm-diagonalinLnk}
The intersection product in theorem \ref{thm-diagonalinRn} can
be rewritten as
$$ \left( \sum_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot
\left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k \cdot
\left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\cdot F^n_n$$
for some Cartier divisors $h_{i,j}$ on $F^n_n$.
\end{theorem}
We have to prepare the proof of the theorem by the following
lemma:
\pagebreak
\begin{lemma} \label{lemma-neededrelations}
Let $C \in Z_l(L^n_{n-k})$ be a subcycle of $L^n_{n-k}$. Then the
following intersection products are zero:
\begin{enumerate}
\item $\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right) \cdot (C \times \RR^n)$ \vspace{2mm},
\item $v_{i_1} \cdots v_{i_{n-k+r}} \cdot (C \times \RR^n)$ \vspace{2mm},
\item $\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^s \cdot v_{i_1} \cdots v_{i_{n-k-s+r}} \cdot (C \times \RR^n),$
\end{enumerate}
where $r,s>0$ and the vectors $$v_{i_j} \in \left\{ \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array}
\right),\ldots,\left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right),
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)\right\}$$ are pairwise distinct.
\end{lemma}
\begin{proof}
(a) and (b): In both cases, a cone that can occur in the
intersection product with non-zero weight has to be contained
in a cone of $F^n_n$ that is contained in $|L^n_{n-k} \times \RR^n|$
and that contains the vectors $\left( \begin{array}{c}
-e_0 \\ \hline 0 \end{array} \right), \left( \begin{array}{c} 0
\\ \hline -e_0 \end{array} \right)$ or $v_{i_1}, \ldots,
v_{i_{n-k+r}}$, respectively. But there are no such cones.\\
(c): By (a) and \cite[lemma 9.7]{AR07} we can rewrite the intersection product as
$$\begin{array}{rl}
& \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^s \cdot v_{i_1} \cdots v_{i_{n-k-s+r}} \cdot (C \times \RR^n)\\
=& \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)+\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^s \cdot v_{i_1} \cdots v_{i_{n-k-s+r}} \cdot (C \times \RR^n)\\
=& \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \cdot v_{i_1} \cdots v_{i_{n-k-s+r}} \cdot
\left[ \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)+\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^s \cdot C \right] \times \RR^n\\
=& \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \cdot v_{i_1} \cdots v_{i_{n-k-s+r}} \cdot
\left[ \max \{0,x_1,\ldots,x_n \}^s \cdot C \right] \times \RR^n,
\end{array}$$
which is zero by (b) as $\max \{0,x_1,\ldots,x_n \}^s \cdot C$ is contained in $L^n_{n-k-s}$.
\end{proof}
\begin{proof}[Proof of theorem \ref{thm-diagonalinLnk}]
By theorem \ref{thm-diagonalinRn} we have the representation
$$\begin{array}{rcl}
\triangle_{L^n_{n-k}} &=& { \tiny \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\dots \left( \left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)} \cdot
\underbrace{ {\tiny \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\right)^k} \cdot [F^n_n]}_{=[L^n_{n-k} \times \RR^n]} \\
&=& { \tiny \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)
\cdots
\left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right)
+ \ldots + \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^n \right) } \cdot
[L^n_{n-k} \times \RR^n].
\end{array}$$
By lemma \ref{lemma-neededrelations} (b) all the summands containing $\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^s$ with a power $s<k$ are zero. Hence we can rewrite the
intersection product as
{\small $$\begin{array}{rcl} \triangle_{L^n_{n-k}} &=& \left[ \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)\cdots
\left( \begin{array}{c} -e_{n-k} \\ \hline 0 \end{array} \right)
+ \ldots + \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^{n-k} \cdot
\left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\right)^k -A \right] \vspace{1mm}\\ & & \cdot [ L^n_{n-k} \times \RR^n],
\end{array}$$}where $A$ contains all the summands we added too much.
Thus all the summands of $A$ are of the form
$$ \alpha \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^t$$
for some integer $\alpha$, vectors $v_{i} \in \left\{ \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right),\dots,
\left( \begin{array}{c} -e_n \\ \hline 0 \end{array}
\right) \right\}$ and powers $1 \leq t \leq k$, ${0 \leq s \leq n}$. By lemma
\ref{lemma-neededrelations} (b) and (c) such a summand applied to $[ L^n_{n-k} \times \RR^n]$ is zero if $s<k$
and only those summands remain in $A$ that have $t \geq 1, s \geq
k$. Let $$ S:=\alpha \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^t$$ be one of the remaining summands. By lemma
\ref{lemma-neededrelations} (a) we obtain the equation
$$\begin{array}{rl}
& \alpha \cdot v_1 \cdots v_{n-s-t} \cdot { \tiny \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^t \cdot [ L^n_{n-k} \times \RR^n] }\\
= & {\tiny \left( \sum\limits_{j=0}^t {{t}\choose{j}} \cdot \alpha \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^j \cdot \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)^{t-j} \right) } \cdot [ L^n_{n-k} \times \RR^n]\\
= & {\tiny \left( \alpha \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s} \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^t \right) } \cdot [ L^n_{n-k} \times \RR^n]\\
= & { \tiny \left[ \left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\right. } \\ & { \tiny \left. \cdot \left( \alpha \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s-k} \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^t \right) -B_S \right] } \cdot [ L^n_{n-k} \times \RR^n],
\end{array}$$
where $B_S$ contains again all the summands we added too much.
Thus all the summands of $B_S$ are of the form
$$ S':=\beta \cdot {{t}\choose{t'}} \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s-s'} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{s'} \cdot \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right)^{t'} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{t-t'}$$
for some integer $\beta$ and powers $1 \leq s' \leq k$, $0 \leq t' \leq t$. If $s-s'<k$ we group all corresponding summands together as
$$ \beta \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s-s'} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{s'} \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^t. $$
This product applied to $[L^n_{n-k} \times \RR^n]$ is zero by lemma \ref{lemma-neededrelations} (b) and
(c). Moreover, all summands $S'$ with $s-s' \geq k$ and $t'>0$
yield zero on $[ L^n_{n-k} \times \RR^n]$ by lemma \ref{lemma-neededrelations} (a). Thus only those summands $S'$ are left in $B_S$ that are of the form
$$ S'=\beta' \cdot v_1 \cdots v_{n-s-t} \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{s-s'} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{t+s'}$$
with $s-s' \geq k$ and $s' \geq 1$. Applying this process inductively to all summands with $t=1,\ldots,n-k-1$ in which we could not factor out
$\left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\right)^k$, yet, we can by and by increase the power of $\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)$
in all remaining summands until finally only one summand
$$ \gamma \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{k} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{n-k} $$
is left. But
$$\begin{array}{l} \gamma \cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right)^{k} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{n-k} \cdot [ L^n_{n-k} \times \RR^n] \\
= \gamma \cdot \left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}
\right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^k \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^{n-k} \cdot [ L^n_{n-k} \times \RR^n] \end{array}$$
as
$$\begin{array}{l} \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^{i} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{k-i} \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array}
\right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right) \right)^{n-k} \cdot [L^n_{n-k} \times \RR^n]\\
=\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^{i} \cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{k-i} \cdot [L^n_{0} \times \RR^n]\\
= 0 \end{array}$$
for all $i<k$ by lemma \ref{lemma-neededrelations} (b) and
$$ \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^{k} \cdot \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right)^{j}
\cdot \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)^{n-k-j} \cdot [L^n_{n-k} \times \RR^n] = 0$$
for all $j>0$ by lemma \ref{lemma-neededrelations} (a). This
proves the claim.
\end{proof}
\begin{example}
We perform the steps described in the proof of theorem
\ref{thm-diagonalinLnk} for the case $n=3, k=2$:\\
By theorem \ref{thm-diagonalinRn} we have the representation:
$$\begin{array}{rcl}
\triangle_{L^3_1} &=& { \tiny \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\cdot \left( \left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\cdot \left( \left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)
\right)} \vspace{2mm}
\\ && \cdot \underbrace{ {\tiny \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\right)^2} \cdot [F^3_3]}_{=[L^3_1 \times \RR^3]} \\
&=& \Big( { \tiny \underbrace{\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)}_{=0 \text{ by lemma \ref{lemma-neededrelations} (b)}}+
\underbrace{\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)}_{=0 \text{ by lemma \ref{lemma-neededrelations} (b)}}} \vspace{2mm}\\
&&+
{ \tiny \underbrace{\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)}_{=0 \text{ by lemma \ref{lemma-neededrelations} (b)}}+
\underbrace{\left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)}_{=0 \text{ by lemma \ref{lemma-neededrelations} (b)}}} \vspace{2mm}\\
&&+
{ \tiny \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^2+
\left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^2+
\left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)
\cdot \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^2+
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^3 } \Big) \vspace{2mm}\\
&& \cdot [L^3_1 \times \RR^3].
\end{array}$$
Now we factor out ${ \tiny \left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)+
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\right)^2}$ and subtract all summands we do not need:
$$\begin{array}{rcl}
\triangle_{L^3_1} &=&
\Big( { \tiny \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)
\bigg) \cdot \bigg( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)+
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\Big)^2} \cdot [L^3_1 \times \RR^3] \vspace{2mm}\\
&& -\Big( { \tiny \underbrace{\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)\left( \begin{array}{c} -e_0 \\ \hline -e_0
\end{array}\right)^2}_{=0 \text{ by \ref{lemma-neededrelations} (b)}}
+\underbrace{\left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)\left( \begin{array}{c} -e_0 \\ \hline -e_0
\end{array}\right)^2}_{=0 \text{ by \ref{lemma-neededrelations} (b)}}
+\underbrace{\left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}\right)^2}_{=0 \text{ by \ref{lemma-neededrelations} (b)}}} \vspace{2mm}\\
&& \tiny{ +\underbrace{\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)\left( \begin{array}{c} -e_0 \\ \hline -e_0
\end{array}\right)^2}_{=0 \text{ by \ref{lemma-neededrelations} (c)}}
+\underbrace{2\left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}\right)\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)}_{=0 \text{ by \ref{lemma-neededrelations} (b)}}
+\underbrace{2\left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}\right)\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)}_{=0 \text{ by \ref{lemma-neededrelations} (b)}} }\vspace{2mm}\\
&& \tiny{ +\underbrace{2\left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array}\right)\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array}
\right)}_{=0 \text{ by \ref{lemma-neededrelations} (b)}}
+2\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^2\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
}\Big) \cdot [L^3_1 \times \RR^3].
\end{array}$$
But by lemma \ref{lemma-neededrelations} (a) and (b) we have the following equation for this last summand:
$$\begin{array}{rl}
& {\tiny -2\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^2\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)}
\cdot [L^3_1 \times \RR^3] \vspace{2mm}\\
=&
{\tiny -2 \left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)^2
+2\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
+\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)^2
\right) \cdot \left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) + \left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)\right)
} \cdot [L^3_1 \times \RR^3].
\end{array}$$
Hence we obtain altogether:
$$\begin{array}{rcl}
\triangle_{L^3_1} &=&
\Big( { \tiny \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} -e_2 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} -e_3 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)-2
\left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right)-2
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\bigg)} \vspace{2mm}\\
&& \cdot {\tiny \bigg( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right)+
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\Big)^2} \cdot {\tiny \bigg( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right)+
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right)
\Big)^2} \cdot [\RR^3 \times \RR^3].
\end{array}$$
\end{example}
\begin{corollary} \label{coro-diagonalinLnk}
The Cartier divisors $h_{i,j}$ from theorem \ref{thm-diagonalinLnk} provide the following description
of the diagonal $\triangle_{L^n_{n-k}}$:
$$\triangle_{L^n_{n-k}} = \sum_{i=1}^r h_{i,1} \dots h_{i,n-k} \cdot
[L^n_{n-k} \times L^n_{n-k}].$$
\end{corollary}
\begin{proof}
Let $x_1,\ldots,x_n$ be the coordinates of the first and $y_1,\ldots,y_n$ be coordinates
of the second factor of $\RR^n \times \RR^n$.
Applying \cite[lemma 9.6]{AR07} we can conclude that
$$\begin{array}{l}
\left[ \left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k \cdot
\left( \left( \begin{array}{c} -e_0 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\cdot F^n_n \right] \vspace{2mm} \\ = \left[ \max \{ 0, x_1,\ldots,x_n \}^{k} \cdot \max \{ 0, y_1,\ldots,y_n \}^{k}
\cdot F^n_n \right] \vspace{2mm} \\ = [L^n_{n-k} \times L^n_{n-k}]
\end{array}$$
and hence by theorem \ref{thm-diagonalinRn} and theorem \ref{thm-diagonalinLnk} that $$\sum_{i=1}^r h_{i,1} \dots h_{i,n-k} \cdot
[L^n_{n-k} \times L^n_{n-k}] = \triangle_{L^n_{n-k}}.$$
\end{proof}
\begin{remark} \label{remark-getdiagonal}
As lemma \ref{lemma-neededrelations} does not only hold on
$L^n_{n-k} \times \RR^n$ but also on any $C \times \RR^n$ with $C$ a
subcycle of $L^n_{n-k}$, the proof of theorem \ref{thm-diagonalinLnk} indeed shows
that
$$\begin{array}{rl}
& \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot
\left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\cdot (C \times \RR^n) \vspace{1mm}\\
= & \left( \left( \begin{array}{c} -e_1 \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\dots \left( \left( \begin{array}{c} -e_n \\ \hline 0 \end{array} \right) +
\left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) \right)
\cdot (C \times \RR^n)\\
\end{array}$$
for all cycles $C \in Z_l(L^n_{n-k})$. Using \cite[corollary 9.8]{AR07} we can conclude that
$$\begin{array}{rl}
& \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot
\left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\cdot (C \times \RR^n) \vspace{1mm}\\
= & \triangle_{\RR^n} \cdot (C \times \RR^n) \vspace{1mm}\\
= & \triangle_{C}
\end{array}$$
for all such cycles $C$.
\end{remark}
\begin{corollary} \label{coro-diagonalinU}
Let $\sigma \in L^n_{n-k}$, let $x \in \sigma$ and let $U \subseteq S_{\sigma}=
\bigcup_{\sigma' \in L^n_{n-k}: \sigma' \supseteq \sigma} (\sigma')^{ri}$
be an open subset of $|L^n_{n-k}|$ containing $x$. Moreover, let $F$ be the open
fan $F:=\{ -x+\sigma \cap U| \sigma \in L^n_{n-k} \}$ and
$\widetilde{F}$ the associated tropical fan. Then there are
Cartier divisors $h'_{i,j}$ on $\widetilde{F} \times \widetilde{F}$ such that
$$\triangle_{[\widetilde{F}]} = \sum_{i=1}^r h'_{i,1} \dots h'_{i,n-k} \cdot
[\widetilde{F} \times \widetilde{F}].$$
\end{corollary}
\begin{proof}
To obtain the Cartier divisors $h'_{i,j}$ we just have to
restrict the Cartier divisors $h_{i,j}$ from corollary \ref{coro-diagonalinLnk}
to the open set $U \times U$, translate them suitably and extend them from $F \times F$ to the
associated tropical fan $\widetilde{F} \times \widetilde{F}$.
\end{proof}
\pagebreak
\begin{example}
The following figure shows two fans associated to open subsets
of $L^3_2$ as in corollary \ref{coro-diagonalinU}:
\begin{center}
\includegraphics[scale=0.66]{pic/CutOpenFans.eps}
\end{center}
\end{example}
\begin{lemma}
Let $C \in Z_k(\RR^n)$ and $D \in Z_l(\RR^n)$ be tropical cycles such that there exist
representations of the diagonals $\triangle_C$ and $\triangle_D$ as sums of products of Cartier
divisors on $C \times C$ and $D \times D$, respectively. Then there also exists a representation of
$\triangle_{C \times D}$ as a sum of products of Cartier divisors on $(C \times D)^2$.
\end{lemma}
\begin{proof}
Let $\triangle_C = \sum_{i=1}^r \varphi_{i,1} \dots \varphi_{i,k} \cdot (C \times
C)$ and $\triangle_D = \sum_{i=1}^s \psi_{i,1} \dots \psi_{i,l} \cdot (D \times
D)$. \linebreak Moreover, let $\pi_{x}, \pi_{y} :(\RR^n )^4 \rightarrow (\RR^n )^2$
be given by $(x_1,y_1,x_2,y_2) \mapsto (x_1,x_2)$ and \linebreak $(x_1,y_1,x_2,y_2) \mapsto
(y_1,y_2)$, respectively.
Then we have the equation
$$\triangle_{C \times D} = \left(\sum_{i=1}^r \pi_{x}^{*}\varphi_{i,1} \dots \pi_{x}^{*}\varphi_{i,k} \right)
\cdot \left(\sum_{i=1}^s \pi_{y}^{*}\psi_{i,1} \dots \pi_{y}^{*}\psi_{i,l} \right)
\cdot (C \times D)^2.$$
\end{proof}
Now we are ready to define intersection products on all
spaces on which we can express the diagonal by means of Cartier
divisors:
\begin{definition}[Intersection products] \label{def-intersectionproduct}
Let $C \in Z_k(\RR^n)$ be a tropical cycle and assume that there are Cartier divisors $\varphi_{i,j}$ on
$C \times C$ such that $$\triangle_{C} = \sum_{i=1}^r \varphi_{i,1} \dots \varphi_{i,k} \cdot
(C \times C).$$
Moreover, let $\pi: C \times C \rightarrow C$ be the morphism given by $(x,y) \mapsto x$.
Then we define the intersection product of subcycles of $C$ by
$$\begin{array}{rcl}
Z_{k-l}(C) \times Z_{k-l'}(C) & \longrightarrow & Z_{k-l-l'}(C)\\
(D_1,D_2) & \longmapsto & D_1 \cdot D_2 := \pi_{*}\left(\sum_{i=1}^r \varphi_{i,1} \dots \varphi_{i,k} \cdot (D_1 \times D_2)
\right).
\end{array}$$
\end{definition}
We use the rest of this section to prove that this intersection
product is independent of the used representation of the diagonal
and that it has all the properties we expect --- at least for
those spaces we are interested in:
\begin{lemma}
Let $C \in Z_k(\RR^n)$ be a tropical cycle,
$D \in Z_{k-l}(C)$, $E \in Z_{k-l'}(C)$ be subcycles, let $\varphi \in \Div(C)$
be a Cartier divisor and ${\pi: C \times C \rightarrow C}$ the morphism
given by $(x,y) \mapsto x$. Then the following equation holds:
$$ (\varphi \cdot D) \times E = \pi^{*}\varphi \cdot (D \times E).$$
\end{lemma}
\begin{proof}
The proof is exactly the same as for \cite[lemma 9.6]{AR07}.
\end{proof}
\begin{corollary} \label{coro-cartdivintprodpermute}
Let $C \in Z_k(\RR^n)$ be a tropical cycle that admits an
intersection product as in definition
\ref{def-intersectionproduct}, let $D \in Z_{k-l}(C)$, $E \in
Z_{k-l'}(C)$ be subcycles and let $\varphi \in \Div(C)$
be a Cartier divisor. Then the following equation holds:
$$ (\varphi \cdot D) \cdot E = \varphi \cdot (D \cdot E).$$
\end{corollary}
\begin{proof}
The proof is exactly the same as for \cite[lemma 9.7]{AR07}.
\end{proof}
\begin{proposition}
Let $D \in Z_{l}(L^n_{n-k})$ be a subcycle. Then the
equation $$ D \cdot [L^n_{n-k}] = [L^n_{n-k}] \cdot D = D$$
holds on $L^n_{n-k}$.
\end{proposition}
\begin{proof}
Let $\pi_i: L^n_{n-k} \times L^n_{n-k} \rightarrow L^n_{n-k}$ be the morphism given by $(x_1,x_2)
\mapsto x_i$. By remark \ref{remark-getdiagonal} we get the equation
$$\begin{array}{rcl}
D \cdot [L^n_{n-k}] &=& (\pi_1)_{*} \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \cdot \left( D \times [L^n_{n-k}] \right) \right)\\
&=& (\pi_1)_{*} \left( \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot
\left( \left( \begin{array}{c} 0 \\ \hline -e_0 \end{array} \right) +
\left( \begin{array}{c} -e_0 \\ \hline -e_0 \end{array} \right) \right)^k
\cdot (D \times \RR^n) \right)\\
&=& (\pi_1)_{*} \left( \triangle_{\RR^n} \cdot (D \times \RR^n) \right)\\
&=& (\pi_1)_{*} \left( \triangle_{D} \right)\\
&=& D.
\end{array}$$
Furthermore, let $\phi:L^n_k \times L^n_k \rightarrow L^n_k \times L^n_k$ be the morphism
induced by $(x,y) \mapsto (y,x)$. Obviously we have the
equality
$$\left( \sum_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot [L^n_{n-k} \times L^n_{n-k}]
= \left( \sum_{i=1}^r \phi^{*}h_{i,1} \dots \phi^{*}h_{i,n-k} \right) \cdot [L^n_{n-k} \times L^n_{n-k}].$$
If $\pi_{ij}: (L^n_{n-k})^4 \rightarrow (L^n_{n-k})^2$
is the morphism given by $(x_1,x_2,x_3,x_4) \mapsto (x_i,x_j)$ and
$$ \triangle := \left(\sum_{i=1}^r \pi_{13}^{*}h_{i,1} \dots \pi_{13}^{*}h_{i,n-k} \right)
\cdot \left(\sum_{i=1}^r \pi_{24}^{*}h_{i,1} \dots \pi_{24}^{*}h_{i,n-k}
\right)$$
we can conclude by \cite[proposition 7.7]{AR07} and \cite[lemma 9.6]{AR07} that
$$ \begin{array}{rl}
& \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot (D \times [L^n_{n-k}])\\
=& \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot
\Big( (D \times [L^n_{n-k}]) \cdot ([L^n_{n-k} \times L^n_{n-k}]) \Big)\\
=& \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot
(\pi_{12})_{*} \Big( \triangle \cdot \left( (D \times [L^n_{n-k}]) \times ([L^n_{n-k} \times L^n_{n-k}]) \right) \Big)\\
=& \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot
(\pi_{12})_{*} \left( \triangle_{D \times [L^n_{n-k}]} \right)\\
=& \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot
(\pi_{34})_{*} \left( \triangle_{D \times [L^n_{n-k}]} \right)\\
=& (\pi_{34})_{*} \left( \left( \sum\limits_{i=1}^r \pi_{34}^{*}\phi^{*} h_{i,1} \dots \pi_{34}^{*}\phi^{*} h_{i,n-k} \right) \cdot
\triangle \cdot \left( (D \times [L^n_{n-k}]) \times ([L^n_{n-k} \times L^n_{n-k}]) \right) \right)\\
=& (\pi_{34})_{*} \left( \triangle \cdot (D \times [L^n_{n-k}]) \times
\left( \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot [L^n_{n-k} \times L^n_{n-k}] \right) \right)\\
=& (\pi_{34})_{*} \left( \triangle \cdot (D \times [L^n_{n-k}]) \times
\left( \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot [L^n_{n-k} \times L^n_{n-k}] \right) \right)\\
=& \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot (D \times [L^n_{n-k}]).
\end{array}$$
Hence we can deduce that
$$\begin{array}{rcl}
D \cdot [L^n_{n-k}] &=& (\pi_1)_{*} \left( \triangle_{D} \right)\\
&=& (\pi_2)_{*} \left( \triangle_{D} \right)\\
&=& (\pi_2)_{*} \left( \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot (D \times [L^n_{n-k}]) \right)\\
&=& (\pi_2)_{*} \left( \left( \sum\limits_{i=1}^r \phi^{*} h_{i,1} \dots \phi^{*} h_{i,n-k} \right) \cdot (D \times [L^n_{n-k}]) \right)\\
&=& (\pi_1)_{*} \left( \left( \sum\limits_{i=1}^r h_{i,1} \dots h_{i,n-k} \right) \cdot ([L^n_{n-k}] \times D) \right)\\
&=& [L^n_{n-k}] \cdot D.
\end{array}$$
This proves the claim.
\end{proof}
\begin{remark}
We can prove in the same way that $[L^n_{n-k} \times L^m_{m-l}] \cdot D = D$
holds for all subcycles $D$ of $L^n_{n-k} \times L^m_{m-l}$ and
even that $[L^{n_1}_{{n_1}-{k_1}} \times \ldots \times L^{n_r}_{{n_r}-{k_r}}] \cdot D = D$
holds for all $r \geq 1$ and all subcycles $D$ of $L^{n_1}_{{n_1}-{k_1}} \times \ldots \times L^{n_r}_{{n_r}-{k_r}}$.
Moreover, restricting the intersection products to open subsets of $|L^n_k|$ or
$|L^{n_1}_{{n_1}-{k_1}} \times \ldots \times L^{n_r}_{{n_r}-{k_r}}|$,
respectively, implies that $X \cdot D = D$ also holds for all subcycles
$D \in Z_{l}(X)$ if $X \in \{ [\widetilde{F}], [\widetilde{F_1} \times \ldots \times \widetilde{F_r}] \}$
where $\widetilde{F}$, $\widetilde{F_i}$ are tropical fans associated to an open
subsets of some $|L^n_k|$ like in corollary \ref{coro-diagonalinU}.
\end{remark}
\begin{proposition}
Let $C \in Z_k(\RR^n)$ be a tropical cycle that admits an
intersection product as in definition \ref{def-intersectionproduct}
and let $D,D' \in Z_l(C)$, $E \in Z_{l'}(C)$ be subcycles. Then the following equation holds:
$$(D+D') \cdot E = D \cdot E+ D' \cdot E.$$
\end{proposition}
\begin{proof}
The proof is exactly the same as for \cite[theorem 9.10 (b)]{AR07}.
\end{proof}
\begin{proposition} \label{propo-independence}
Let $C \in Z_k(\RR^n)$ be a tropical cycle that admits an
intersection product as in definition \ref{def-intersectionproduct}
and let $D \in Z_l(C)$ be a subcycle of $C$. Moreover, let $E \in Z_{l'}(C)$
be a subcycle such that there are Cartier divisors $\psi_{i,j} \in \Div(C)$
with $$\sum_{i=1}^r \psi_{i,1} \dots \psi_{i,k-l'} \cdot C = E.$$
If additionally $C \cdot D = D$ holds then $$\sum_{i=1}^r \psi_{i,1} \dots
\psi_{i,k-l'} \cdot D = E \cdot D.$$
\end{proposition}
\begin{proof}
The proof is the same as for \cite[corollary 9.8]{AR07}.
\end{proof}
\begin{remark} \label{remark-independence}
The meaning of proposition \ref{propo-independence} is the following: If $X \in Z_k(\RR^n)$
is a tropical cycle such that the diagonal $\triangle_{X}$ can be written
as a sum of products of Cartier divisors as in definition \ref{def-intersectionproduct}
and additionally $(X \times X) \cdot Y = Y$ is fulfilled for all subcycles
$Y$ of $X \times X$ then we can apply proposition \ref{propo-independence} with
$C:=X \times X$ and $E:=\triangle_X$ to deduce that the definition of the intersection product is
independent of the choice of the Cartier divisors describing the
diagonal. In particular we have well-defined intersection products on
$L^n_k$, $L^{n_1}_{k_1} \times \ldots \times L^{n_r}_{k_r}$, $\widetilde{F}$ and
$\widetilde{F_1} \times \ldots \times \widetilde{F_r}$
for all tropical fans $\widetilde{F}$, $\widetilde{F_i}$ associated to an open subset
of some $|L^n_k|$ like in corollary \ref{coro-diagonalinU}.
\end{remark}
\begin{theorem} \label{theorem-intproductproperties}
Let $C \in Z_k(\RR^n)$ be a tropical cycle that admits an
intersection product as in definition \ref{def-intersectionproduct}
such that additionally $(C \times C) \cdot D = D$ is fulfilled
for all subcycles $D$ of $C \times C$.
Moreover, let $E,E' \in Z_l(C)$, $F \in Z_{l'}(C)$ and
$G \in Z_{l''}(C)$ be subcycles. Then the following equations hold:
\begin{enumerate}
\item $E \cdot F = F \cdot E$,
\item $(E \cdot F) \cdot G = E \cdot (F \cdot G).$
\end{enumerate}
\end{theorem}
\begin{proof}
The proof is exactly the same as for \cite[theorem 9.10 (a) and (c)]{AR07}.
\end{proof}
We finish this section with an example showing that even curves
intersecting in the expected dimension can have negative
intersections:
\begin{example}
Let $C, D \in Z_1(L^3_2)$ be the curves shown in the figure. We
want to compute the intersection $C \cdot D$. By proposition
\ref{propo-independence} the easiest way to achieve this is to
write one of the curves as $\psi \cdot [L^3_2]$ for some Cartier divisor
$\psi$ on $L^3_2$.
\begin{center}
\input{pic/curves_in_plane}
\end{center}
Let $F$ be the refinement of $L^3_2$ arising by dividing the cones $\langle -e_1,-e_2 \rangle_{\RR_{\geq 0}}$
and $\langle -e_0,-e_3 \rangle_{\RR_{\geq 0}}$ into cones $\langle -e_1,-e_1-e_2 \rangle_{\RR_{\geq 0}}$,
$\langle -e_2,-e_1-e_2 \rangle_{\RR_{\geq 0}}$ and $\langle -e_0,-e_0-e_3 \rangle_{\RR_{\geq 0}}$,
${\langle -e_3,-e_0-e_3 \rangle_{\RR_{\geq 0}}}$, respectively. Then
$$\psi:=\left(\begin{array}{c}1\\1\\1\end{array}\right)-\left(\begin{array}{c}-1\\-1\\0\end{array}\right)$$
defines a rational function on $F$. As shown in \cite[example 3.10]{AR07}
we have $\psi \cdot [L^3_2] = C$. Hence we can calculate
$$\begin{array}{rcl}
C \cdot D = \psi \cdot D &=& \left( \psi \left(\begin{array}{c}-2\\-3\\0\end{array}\right) + \psi \left(\begin{array}{c}2\\2\\1\end{array}\right)
+ \psi \left(\begin{array}{c}0\\1\\1\end{array}\right) - \psi \left(\begin{array}{c}0\\0\\0\end{array}\right) \right) \cdot \{0\}\\
&=& (-2+0+1-0) \cdot \{0\}\\
&=& -1 \cdot \{0\}.
\end{array}$$
\end{example}
\begin{remark}
This result is remarkable for the following reason: Our ambient space
$L^3_2$ arises as a so-called \emph{modification} of $\RR^2$ (cf. \cite{M06}, \cite{M07}).
Varieties that are connected by a series of modifications are
called \emph{equivalent} by G. Mikhalkin and are expected to
have similar properties. But the above example shows that there
is a big difference between $\RR^2$ and $L^3_2$ even though they
are equivalent: On $\RR^2$ there is no negative intersection
product of curves, on $L^3_2$ there is.
\end{remark}
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TITLE: Among triangles of perimeter $3a$ and a side $a$, what is the probability of selecting an acute/right/obtuse/scalene/isosceles triangle?
QUESTION [2 upvotes]: If I consider all possible triangles with perimeter equal to $3a$ and one side length equal to $a$, what is the probability of selecting an acute angled triangle? a right triangle? an obtuse angled triangle? a scalene triangle? an isosceles triangle?
How would I approach such a problem mathematically? (I know writing a simulation on a computer is quite easy.)
Please share if you have a reference that discusses how to solve problems like this. I am a math hobbyist at best, not a mathematician, so would appreciate something readable. Thank you.
REPLY [0 votes]: Another option is to look at the path
of all such triangles through the "map of Trilandia".
Let $\rho,r,R$ denote the semiperimeter,
inradius and the circumradius of a general triangle.
Its shape is uniquely defined by the pair
$v=\rho/R, v=r/R$.
Consider all possible shapes of triangles
inscribed into the circle with $R=1$.
Any particular shape is represented by the point $(v,u)$
on the map:
For $v\in[0,\tfrac12]$
the boundaries are defined by two curves,
$u_{\min}$ (blue) and $u_{\max}$ (red)
\begin{align}
u_{\min}&=\sqrt{27-(5-v)^2-2\sqrt{(1-2\,v)^3}}
\tag{1}\label{1}
,\\
u_{\max}&=\sqrt{27-(5-v)^2+2\sqrt{(1-2\,v)^3}}
\tag{2}\label{2}
\end{align}
The point $E$ corresponds to the equilateral shape.
The orange line $u_{90}=v+2$ corresponds
to all triangles with $90^\circ$ angle,
and separates the areas of acute (top)
and obtuse (bottom) kingdoms,
so
the top border represents exclusively acute isosceles shapes,
while the bottom border line ($u_{\min}$) is split
by the line $u_{90}=v+2$
at the check-point $(\sqrt2-1,\sqrt2+1)$
between the lower obtuse isosceles part
and the short upper isosceles part.
Note that part $BD$ of
the orange line $u_{90}$ escapes the Trilandia,
that means that for $v>\sqrt2-1$
no valid right triangle can be constructed.
The green curve represent all possible triangular shapes
inscribed in a unit circle,
for which one of the side lengths is $2\rho/3$.
To find the equation of the green line, recall that
the three side length of the triangle, inscribed into unit circle
are
the roots of cubic equation
\begin{align}
x^3-2u\,x^2+(u^2+v^2+4v)\,x-4\,u\,v&=0
\tag{3}\label{3}
,
\end{align}
given that one side length is equal $\tfrac23\,u$,
as a reminder
of \eqref{3} divided by $x-\tfrac23\,u$,
we have a condition
\begin{align}
u^2+9\,v^2-18\,v&=0
\tag{4}\label{4}
,
\end{align}
so the equation of the sought line is
\begin{align}
u(v)&=3\sqrt{2v-v^2}
\tag{5}\label{5}
,\\
u'(v)&=
\frac{3(1-v)}{\sqrt{v\,(2-v)}}
.
\end{align}
It crosses the
obtuse/acute
border at the point $(\tfrac25,\tfrac{12}5)$.
So, if we assume that probabilities are proportional to the length of the curve,
we have
the total length of the curve of interest as
\begin{align}
L&=
\int_0^{1/2}
\sqrt{1+(u'(v))^2}
\, dv
\approx 2.670
,
\end{align}
and the length of the obtuse part
\begin{align}
L_o&=
\int_0^{2/5}
\sqrt{1+(u'(v))^2}
\, dv
\approx 2.449
,
\end{align}
so the probability of choosing an obtuse shape in this case is
\begin{align}
P_o&=\frac{L_o}{L}
\approx 91.7\%
,
\end{align}
which leaves a pity $8.3\%$
for acute shapes.
| 196,637
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Action Photo Practice
You should have 10-12 SUCCESSFUL panning style pictures using shutter speeds of around 1/20-1/60 and then 10-12 pictures of regular fast shutter speeds (usually 1/250-1/2000).
Have your subjects repeat the same action over and over.
How many of you remembered to bring some sporting equipment?
Suggested activities to shoot: someone running across the field, someone doing back handsprings or other gymnastic moves, anyone playing catch with a football or baseball, and hackysack etc.
Crocodile Hunter.
Please return to class at 11:35 so you can upload your pictures and turn in the Contact Sheets.
Your SC Sports pictures are due Tuesday.
We have a quiz Monday.
1 comment:
Preheat oven to 400 F. Finally most dish washers will clean your dishes period, but one needs to be on the look out for models that have extra features inline with how you like your dishes done. The humidifier quietly heats the water to soothe the throat and make your breathing feel better again. There are a number of companies in your area that employ experienced and educated professionals that can help with water heater repair. Average families go through three sets of dishes daily; one for breakfast, one for lunch, and one for dinner.
| 144,679
|
stamina. As you attach significance to the activities you carry out on the site, context emerges. Your focus sharpens, your intuition takes over, and you gain receptivity towards possibilities. To help get you that level, I have developed a system of LinkedIn absolutes.
Since I started my LinkedIn consultancy in early 2007, I have zeroed in on the manner in which businesspeople have responded to the demands of a disruptive technology. I have fashioned my observations into a unique methodology in teaching LinkedIn success principles—integrating traditional marketing techniques where applicable—while focusing on behaviors that create positive change.
The basic idea of LinkedIn is easier said than done: First, create a magnificent profile that illuminates you, conveys your professional value, and encourages others to connect with you. Next, maneuver about the site in a manner that captures attention and fosters relationship building. Finally, convert your experiences on LinkedIn into tangible, real-world business results. Simple, yes?
LinkedIn, in all its various aspects, is a lesson in cause and effect. Each action or inaction has consequences. Through my work, which has long since been my passion, I have developed what I call the Absolutes of LinkedIn, an educational foundation that will help motivated, opportunity-oriented professionals build eminence, gain a competitive advantage, and maximize their potential on LinkedIn.
1). Visualize a Desired Outcome
Identify a desired outcome (e.g., new client, new job, a good connection, etc.) and see it vividly in your mind’s eye. That which follows now has order and structure. Design your LinkedIn profile around that desired outcome. Assemble your LinkedIn network around that desired outcome. Initiate or advance each conversation around that desired outcome.
Envision the transaction, the passing of the check to you. Hear the voice of the hiring manager offering you the position. Feel the firmness of the handshake upon meeting the decision maker for the first time.
♦ Successful use of LinkedIn is a function of mindset. Without the proper mindset, your work on LinkedIn will be mechanical, detached, and misguided. Also, never underestimate the power of positive psychology. Optimism opens doors and generates opportunities.
2). Crystallize your Message
Clarity is king. Be clear on what you wish to accomplish professionally. Deconstruct your message into its individual components (the what, the how, the when, the where, and the why), and evaluate each piece on its stand-alone merits. Next, specify the situations in which you excel and the types of people or organizations for whom you create value. Coalesce these pieces into a story, one that you own, one for which you wish to become known, and one that sells you.
♦ If you are a problem solver, mention the types of problems that you solve; if you are a trusted advisor, explain how others benefit from your counsel; if you are a value creator, reveal how you improve the lives of others; if you are in or looking for a leadership role, discuss what fuels your passion to serve others.
♦ Razor-sharp branding is essential on LinkedIn. In a distracted society, you only get a precious few seconds to command someone’s attention, let alone make an impact. If your message is on-target and uniquely expressed, you will facilitate greater ease of recall on you, your company, and your offerings.
♦ Any hint of ambiguity or vagueness in your LinkedIn profile will diminish your value in the eyes of a potential client, strategic alliance, or employer, and render you unmemorable. Choose to become indelible.
3). Idealize your LinkedIn Profile
Commit and conform to an ideal or “best version” of yourself. A strong self-concept is essential in leveraging LinkedIn to achieve your desired outcome. The purpose of your LinkedIn profile is to convey your professional value in a manner that drives a favorable first (and lasting) impression. Fearlessly guide the perceptions of visitors. The voice that comes through the writing, in combination with the visual elements on the page, must engender trust and shape the attitudes, judgments, and beliefs of others. What must those you wish to serve know about you?
♦ When you elevate your self-image accordingly, you can set realistic expectations for your use of LinkedIn. With increased confidence comes more concentrated activity, which translates into tangible results.
♦ Your LinkedIn profile gives rise to a social brand which accompanies you in the real world. A profile that fails to capture the essence of the subject falls flat and neutralizes any attempt to gain traction on the site.
4). Emphasize your Value to the LinkedIn Community
Think of your mission on LinkedIn in terms of running an ongoing credibility campaign. LinkedIn provides the infrastructure through which you can accomplish multiple marketing objectives. To be regarded by your target audience, your content must be sharp, relevant, and biased toward you. The more individual expression you put forth on LinkedIn, the more likely you will cross paths with the people you need to meet. Keep telling us what it is you bring to the table and why we should offer you a seat.
♦ On your LinkedIn profile, you create the business case for yourself. Establish your page as a research platform so that your transaction partners, would-be connections, or recruiters can learn about you. Provide the requisite inbound and outbound links.
♦ Take creative liberties with your brand story through video pieces, reinforce your value through published long-form posts (LinkedIn blogs), and demonstrate social proof through the well-crafted LinkedIn recommendations of those who bear witness to your skills and expertise.
♦ All roads lead to and from your LinkedIn profile. Help the right people get there.
5). Personalize all LinkedIn Correspondence
Inject your personality into each piece of LinkedIn communication. Do not be lackadaisical in messaging or approaching others; this is formal, person-to-person business communication that can make or break you. Reach out to people on a human level and you will be seen as someone who cares about developing professional relationships. LinkedIn is not a site on which to blast emails. Not a soul in your network will appreciate a message that begins with “Dear Valued Connection.”
When inviting someone to connect on LinkedIn, do not send the default, standardized greeting. Serve reminder as to where, when, and how you met, and why you feel connecting is a good idea. The same holds true for recommendation requests, extending congratulations to a connection for landing a new position, or celebratory best wishes to a connection on a birthday or work anniversary.
♦ In all cases, delete the prefab text. LinkedIn master communicators do not work off templates; rather, they appeal to the sensibilities of the recipient with a personal message, one that is composed to fit the situation and designed to evoke a response.
♦ Show respect to your fellow LinkedIn citizens and it will be returned to you—in spades.
6). Prioritize your LinkedIn Network
Forging new inroads to meaningful conversations is the key to progress on LinkedIn. Business or career opportunities arise when you organically assemble and effectively manage an engaged community of professionals who are both emotionally and strategically aligned with you. When you tap the social capital of your LinkedIn network, certain connections will emerge as having potential economic value to you, and you to them. These are the relationships to which to attach priority.
♦ Identify those people who can be instrumental in moving you toward your desired outcome and focus on developing those relationships. Give special consideration to those first-degree connections who have history with you and understand the nature of your business.
♦ Over time, you have likely moved within a number of professional circles and acquired a certain status in the eyes of your peers. Source long-time LinkedIn connections from various organizations to which you belong, your alma mater(s), and events you attended. It may have been a while since you chatted with close colleagues, former associates, or classmates. Reach out, renew, and refresh. Circumstances do change. Perhaps there are ways in which you can now help each other.
♦ When mutual benefit is involved, the priority of an engagement shifts, and conversation flows from a place of trust.
7). Characterize your LinkedIn Interactions
Your actions, as well as your words, illuminate your qualities and differentiate you on LinkedIn. Having a formidable presence on the site translates to being immediately recognizable. To that end, look for ways to distinguish yourself through your interactions. Approach each LinkedIn exchange with the goal of providing some insight into the qualities and traits that you possess. Reveal your giving nature, desire to help, and availability to mentor. Show a bit of humility, perhaps a touch of humor. When you are being authentic and intentional, people notice, and they will widen the window into their world for you.
♦ Abide by the laws of LinkedIn etiquette and uphold the standards of quality on the site. Assign professional ethics to how you “like,” comment on, and share the content of others. Approach those who have viewed your LinkedIn profile with sincerity and, when it makes sense, encourage them to connect.
♦ Endorse your connections for skills or expertise only when they have demonstrated them to you—and without expectation of something in return. Craft strong LinkedIn recommendations for those connections who truly deserve them, and do not request recommendations unless you have earned them.
♦ Do not join LinkedIn groups around which your ideal prospects cluster then ram a sales pitch down their throats. These are politically-correct forums that are moderated for quality control, and on which you will be lauded for observing the rules and respecting your fellow group members.
8). Capitalize on your Conversations
Pull potentially meaningful conversations off of LinkedIn as quickly as possible. LinkedIn becomes real at the point when you “break the plane” between the virtual and the physical worlds. Be ready to assume real-world accountability for the professional relationship that you are looking to build. Having that opening dialog is one thing; sustaining it is an altogether different (and more difficult) proposition.
Business conversations generated on LinkedIn have a shelf-life; some end at the point of invitation while others pick up steam. With each new first-degree connection, follow up with a thank-you note. If you perceive that there are synergies worth exploring, you be the proactive party and keep the momentum going.
♦ Keep meaningful conversations in play. Match each touch point received with one of your own. Granted, not everyone with whom you interact on LinkedIn will be receptive to your business overtures. But failure to follow up with those who may prove to be valuable professional resources to you can be costly.
9). Mobilize!
Nothing happens on (or off of) LinkedIn until you activate. LinkedIn is a meritocracy; successful use of the platform is tied to individual ability. In business, inertia is the antithesis of growth. If you wish to generate economic opportunities through your use of LinkedIn, you must get moving and stay moving. Do not be tentative. Put in the time. Learn the site. Glide through each user session with the goal of making something good happen (e.g., gain new industry insights, connect with a motivated prospect, receive an introduction to a decision maker). When there is something to gain, the work becomes fun.
Newton’s First Law of Motion states that a body at rest will remain at rest unless an outside force acts on it, and a body in motion at a constant velocity will remain in motion in a straight line unless acted upon by an external force.
♦ My approach to LinkedIn is rooted in best practices—those which have been shown to drive superior results—and tied to a personal philosophy of representing myself well, connecting judiciously, and respectfully advancing business conversations. I engage on the site with purpose, always looking to bring people of good character into my professional network.
♦ Remember: LinkedIn is an input-equals-output proposition.
Be that body in motion that stays in motion.
[Impetus for this post comes from the late Charley Lau, a hitting coach in professional baseball who was revered for his extraordinary ability to teach the art of hitting on the Major League level. Lau consolidated his instructional framework into “The Absolutes of Hitting,” a manifesto which emphasizes the convergence of the hands, body, and mind in authoritatively hitting a baseball.] featured blogger.
| 277,849
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4:32 pm
Fri April 18, 2014
Kentucky Natural Lands Continues Buying Heavily-Forested Land to Restore, Connect Wild Lands
As this year's observance of Earth Day approaches, the Kentucky Natural Lands Trust continues to buy up property in typically heavily-forested areas. It's a statewide land trust with a mission to protect, restore, and connect wild lands.
KNLT Director Hugh Archer admits it's costly to purchase land, but it is a conservation measure.
"We are a mineral resources removal based economy for 200 years,” he said. “There's a lot of inertia against anything but private property ownership. So, it's an uphill fight here in Kentucky. We own less of our own state than all the other states in the southeast for public purposes."
One of the current major conservation projects for the Kentucky Natural Lands Trust involves land in the Pine Mountain area.
| 45,812
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