Unbounded regions of Infinitely Logconcave Sequences

8.1 Definition .

A duplicial algebra over $K$ is a vector space $A$ equipped with two bilinear maps $/$ , $\backslash:A\otimes A\longrightarrow A$ , verifying the following relations:

x/(y/z)=(x/y)/z\hfill\default@cr x/(y\backslash z)=(x/y)\backslash z\hfill% \default@cr x\backslash(y\backslash z)=(x\backslash y)\backslash z,\hfill\default@cr

for $x,y,z\in A$ .

Definition 4 .

A bi-invariant metric on $\mathcal{G}$ is a distance $d$ on $\mathcal{G}$ such that for any $\phi,\psi,\chi$ in $\mathcal{G}$ ,

d(\phi,\psi)=d(\phi\chi,\psi\chi)=d(\chi\phi,\chi\psi).

It will be said $C^{-1}$ if its composition with the map $\Phi:H\mapsto\phi_{H}^{1}$ is a continuous map $\Phi^{-1}(\mathcal{G})\times\Phi^{-1}(\mathcal{G})\to\mathbb{R}$ , where $\Phi^{-1}(\mathcal{G})\subset Ham$ is endowed with the compact-open topology.

Definition 2.6

A linear transformation $d$ of a Leibniz algebra $L$ is called a derivation if for any $x,y\in L$

d([x,y])=[d(x),y]+[x,d(y)].

Definition 1.1 .

A multiplicative Lie algebra consists of a multiplicative (possibly nonabelian) group $L$ together with a binary function $\{\;,\;\}:L\times L\to L$ , which we shall call Lie product, satisfying the following identities for all $x,x^{\prime},y,y^{\prime},z$ in $L$

(1.1)		$\displaystyle\{x,x\}=1\;,$
(1.2)		$\displaystyle\{x,yy^{\prime}\}=\{x,y\}\;^{y}{\{x,y^{\prime}\}}\;,$
(1.3)		$\displaystyle\{xx^{\prime},y\}=\,^{x}{\{x^{\prime},y\}}\{x,y\}\;,$
(1.4)		$\displaystyle\{\{y,x\},^{x}z\}\{\{x,z\},^{z}y\}\{\{z,y\},^{y}x\}=1\;,$
(1.5)		${}^{z}{\{x,y\}}=\{^{z}x,^{z}y\}\;.$

Definition 3.1 .

A map $f:X\to Y$ between metric spaces has metrically parallel fibers if for every $x,y\in X$ there exists $z\in X$ such that

d(x,z)=d(f(x),f(z))\text{ and }f(y)=f(z).

Definition 6.2 .

Let $\sigma$ and $\tau$ be elements of $\Gamma^{*}$ . Then we define

(\sigma,\tau)=\langle\sigma dx,\tau dy\rangle.

Definition 3.1 .

A Riemannian manifold $(M^{n+1},g)$ is asymptotically hyperbolic provided it is conformally compact, with smooth conformal compactification $(\tilde{M},\tilde{g})$ , and with conformal boundary $\partial\tilde{M}=S^{n}$ , such that the metric $g$ on a deleted neighborhood $(0,T)\times S^{n}$ of $\partial\tilde{M}=\{t=0\}$ takes the form

g=\sinh^{-2}(t)(dt^{2}+h)\,,

(3.19)

where $h=h(t,\cdot)$ is a family of metrics on $S^{n}$ , depending smoothly on $t\in[0,T)$ , of the form,

h=h_{0}+t^{n+1}k+O(t^{n+2})\,,

(3.20)

where $h_{0}$ is the standard metric on $S^{n}$ , and $k$ is a symmetric $2$ -tensor on $S^{n}$ .

Definition 2.4 .

Let $G$ be a locally finite graph, $F$ a finite graph, and $\phi$ an Aut( $F$ )-voltage assignment on $G$ . We define a graph bundle $G\times^{\phi}F$ to be the graph with vertex set $V(G)\times V(F)$ , with two vertices $(u,i),(v,j)\in G\times^{\phi}F$ adjacent if either one of the following two conditions hold:

(1)

$u\sim v\text{ and }j=i^{\phi(uv)}$
(2)

$u=v\text{ and }i\sim j.$

Definition 3

Let $M$ be a finite dimensional free module over a commutative ring with identity $R$ and let $\langle,\rangle:M\times M\to R$ be an antisymmetric bilinear form such that $\langle\mathbf{x},\mathbf{x}\rangle=0$ for all $\mathbf{x}\in M$ . Then $M$ is a quandle with quandle operation

\mathbf{x}\triangleright\mathbf{y}=\mathbf{x}+\langle\mathbf{x},\mathbf{y}% \rangle\mathbf{y}.

The dual quandle operation is given by

\mathbf{x}\triangleright^{-1}\mathbf{y}=\mathbf{x}-\langle\mathbf{x},\mathbf{y% }\rangle\mathbf{y}.

If $R$ is a field and the form is non-degenerate, i.e., if $\langle\mathbf{x},\mathbf{y}\rangle=\mathbf{0}$ for all $\mathbf{y}\in M$ implies $\mathbf{x}=\mathbf{0}\in M$ , then $M$ is symplectic vector space and $\langle,\rangle$ is a symplectic form ; thus it is natural to refer to such $M$ as symplectic quandles . For simplicity, we will use the term “symplectic quandle over $R$ ” to refer to the general case where $R$ is any ring and $\langle,\rangle$ is any antisymmetric bilinear form. If $\langle,\rangle$ is non-degenerate, we will say $(M,\triangleright)$ is a non-degenerate symplectic quandle over $R$ . $M$ and $M^{\prime}$ are isometric if there is an $R$ -module isomorphism $\phi:M\to M^{\prime}$ which preserves the bilinear form $\langle,\rangle$ .

Definition 5

We say that $\phi,\psi\in H_{m}$ are skew-orthogonal, denoted $\phi\perp\psi$ , if

\langle\phi,\psi\rangle=0.

Definition 3.2 .

Let $(\mathcal{Q},\mathcal{P})=\mathcal{Q}(\mathscr{A})$ and let $\Phi:\mathscr{A}\to\mathcal{Q}$ be the quotient map. Suppose that

\Phi(G)=\Phi(H)\Longrightarrow\mathscr{G}(G)=\mathscr{G}(H)\qquad\textrm{for % all }G,H\in\mathscr{A}.

Then we say that $\Phi$ is faithful . If in addition $(\mathcal{Q},\mathcal{P})$ is normal, then we say that $\Phi$ is faithfully normal .

Definition 2.9 .

Let ${\boldsymbol{\lambda}}$ be a bipartition of rank $n$ . A standard bitableau of shape ${\boldsymbol{\lambda}}$ is a sequence of bipartitions

{\boldsymbol{\emptyset}}={\boldsymbol{\lambda}}[0]\subseteq{\boldsymbol{% \lambda}}[1]\subseteq\cdots\subseteq{\boldsymbol{\lambda}}[n]={\boldsymbol{% \lambda}}

such that the rank of ${\boldsymbol{\lambda}}[k]$ is $k$ , for $0\leqslant k\leqslant n$ . Let ${\bf t}$ be a standard bitableau of shape ${\boldsymbol{\lambda}}$ . Then the residue sequence of ${\bf t}$ is the sequence

(\operatorname{res}(\gamma[1]),\dots,\operatorname{res}(\gamma[n]))\in({% \mathbb{Z}}/e{\mathbb{Z}})^{n}

where $\gamma[k]={\boldsymbol{\lambda}}[k]/{\boldsymbol{\lambda}}[k-1]$ , for $1\leqslant k\leqslant n$ .

Definition 3

Given any states $x,y\in X$ , we shall define their similarity $p(x,y)$ by

p(x,y)=a^{2}(x,y).