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TITLE: Prove that I -xx* is singular if and only x*x = 1
QUESTION [0 upvotes]: (=>) Suppose I - xx* is singular if and only there is a y such that (I−xx*)y=0, i.e. xx* y=y. Now set λ=x* y. Then y=λx, i.e. xx* λx=λx
Thus λx(x* x) = λx => x* x = 1
(<=) Suppose x*x = 1
Please help
REPLY [1 votes]: Just start from the basics. $I-xx^*$ is singular if and only there is a $y$ such that $(I-xx^*)y=0$, i.e. such that $xx^*y=y$. Now set $\lambda=x^*y$. Then $y=\lambda x$, i.e. $xx^*\lambda x=\lambda x$, and you have almost proved one direction.
Make sure that you are aware at each step which expressions are scalars. | 203,149 |
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5 comments ↓
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frompo.com | 152,081 |
\begin{document}
\begin{frontmatter}
\author{Mahouton Norbert Hounkonnou\corref{cor1}${}^1$}
\cortext[cor1]{[email protected]}
\author{Sama Arjika\corref{cor2}${}^1$}
\cortext[cor2]{[email protected]}
\author{ Won Sang Chung\corref{cor3}${}^2$ }
\cortext[cor3]{[email protected]}
\title{\bf New families of
$q$ and $(q;p)-$Hermite polynomials
}
\address{${}^1$International Chair of Mathematical Physics
and Applications \\
(ICMPA-UNESCO Chair), University of
Abomey-Calavi,\\
072 B. P.: 50 Cotonou, Republic of Benin,\\
${}^2$Department of Physics and Research Institute of Natural Science, \\
College of Natural Science, \\
Gyeongsang National University, Jinju 660-701, Korea
}
\begin{abstract}
In this paper, we construct
a new family of $q-$Hermite polynomials denoted by
$H_n(x,s|q).$ Main properties and relations
are
established and proved.
In addition, is deduced a sequence of novel polynomials,
$\mathcal{L}_n(\cdot,\cdot |q),$ which appear to be connected with well known $(q,n)-$exponential functions $E_{q,n}(\cdot)$\, introduced by Ernst in his work entitled: {\it A New Method for $q-$calculus,} (Uppsala Dissertations in
Mathematics, Vol. {\bf 25}, 2002). Relevant results spread in the literature
are retrieved as particular cases. Fourier integral transforms are explicitly computed and discussed. A $(q;p)-$extension of the $H_n(x,s|q)$ is also provided.
\end{abstract}
\begin{keyword}
Hermite polynomials, $q-$Hermite polynomials, generating function, $q-$derivative, inversion formula, Fourier integral transform
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{int1}
The classical orthogonal polynomials and the quantum orthogonal polynomials,
also called $q-$orthogonal polynomials, constitute an interesting set of
special functions.
Each family of these polynomials occupies different levels
within the so-called Askey-Wilson scheme (Askey and Wilson, 1985; Koekoek and Swarttouw, 1998;
Lesky, 2005; Koekoek et {\it al}, 2010).
In this scheme, the Hermite polynomials
$H_n(x)$ are the ground level and are characterized
by a set of properties:
(i) they are solutions of a hypergeometric second order differential equation,
(ii) they are generated by a recursion relation,
(iii) they are orthogonal with respect to a weight function and
(iv) they obey the Rodrigues-type formula.
Therefore, there are many ways to construct the Hermite
polynomials. However, they are more commonly
deduced from their
generating function, i.e.,
\be
\sum_{n=0}^\infty\frac{{\bf H}_n(x)}{n!}t^n=e^{2xt-t^2}
\ee
giving rise to the so-called {\it physicists} Hermite polynomials \cite{Johann Cigler}.
Another family of Hermite polynomials, called
the {\it probabilists} Hermite polynomials, is defined as \cite{Johann Cigler}
\be
\sum_{n=0}^\infty\frac{H_n(x)}{n!}t^n=e^{xt-\frac{t^2}{2}}.
\ee
The Hermite polynomials are at the bottom of a
large class of hypergeometric polynomials
to which most of their properties can be
generalized \cite{Bukweli&Hounkonnou12a}, \cite{Dattoli}-\cite{Habibullah}.
In \cite{Johann Cigler}, Cigler introduced
another family of Hermite polynomials $H_n(x,s)$ generalizing
the {\it physicists} and {\it probabilists} Hermite polynomials as
\be
\label{samamasaaa}
\sum_{n=0}^\infty\frac{H_n(x,s)}{n!}t^n=e^{xt-s\frac{t^2}{2}}
\ee
with $H_n(x,1)=H_n(x)$ and $H_n(2x,2)={\bf H}_n(x).$
In this work, we deal with a construction of two new families of $q$ and $(q;p)-$Hermite
polynomials.
The paper is organized as follows. In Section \ref{sction1}, we
give a quick overview on
the Hermite polynomials $H_n(x,s)$ introduced in \cite{Johann Cigler}.
Section \ref{sction2} is devoted to the construction of
a new family of $q-$Hermite polynomials $H_n(x,s|q)$
generalizing the
discrete $q-$Hermite polynomials. The
inversion formula and relevant properties of these polynomials are computed and discussed. Their Fourier integral
transforms are performed in the Section \ref{sction3}.
Doubly indexed Hermite polynomials and some concluding remarks are introduced in Section \ref{sction4}.
\section{On the Hermite polynomials $H_n(x,s)$}
\label{sction1}
In \cite{Johann Cigler}, Cigler showed that the Hermite polynomials $H_n(x,s)$ satisfy
\be
\label{4Hermdiff}
DH_n(x,s)= n\, H_{n-1}(x,s)
\ee
and
the
three term recursion relation
\be\label{4Hermttr}
H_{n+1}(x,s)=x\, H_n(x,s)-s\,n \,H_{n-1}(x,s),\quad n\geq 1
\ee
with $H_0(x,s):=1.\;D:=d/dx$ is the usual differential operator. Immediatly, one can see that
\be
\label{sama:initial}
H_{2n}(0,s)=(-s)^{n}\prod_{k=1}^n(2k-1),\quad H_{2n+1}(0,s)=0.
\ee
The computation of the first fourth polynomials gives:
\be
H_{1}(x,s)= x,\;
H_{2}(x,s)=x^2-s,\;
H_{3}(x,s)=x^3-3\,s\,x,\;
H_{4}(x,s)=x^4-6\,s\,x^2+3\,s^2.
\ee
More generally, the explicit formula of $H_{n}(x,s)$ is written as
\cite{Johann Cigler}
\be
\label{sasama}
H_{n}(x,s)=n!\sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{ (-1)^k\,s^k }{ (2k)!! } \frac{ x^{n-2k} }{(n-2k)! }
=x^n\,{}_2F_0\Bigg(\begin{array}{c}-\frac{n}{2},\frac{1-n}{2}
\\-\end{array}\Big|\;-\frac{2s}{x^2}\Bigg),
\ee
where $({}^n_k)=n!/k!(n-k)!$ is a binomial coefficient,
$n!:=n(n-1)\cdots 2\cdot1,\; (2n)!!:=2n(2n-2)\cdots 2.$\\\\
The symbol $\lfloor\,x\,\rfloor$ denotes the greatest integer
in $x$ and ${}_2F_0$ is called the hypergeometric
series \cite{ASK}.
From
(\ref{4Hermdiff}) and (\ref{4Hermttr}), we have
\be
H_{n }(x,s)= (x-sD)\,H_{n-1 }(x,s),
\ee
where the operator $x-sD $ can be expressed as
\cite{Johann Cigler}
\be
x-s D = e^{\frac{x^2}{2s}} (-sD)\,e^{-\frac{x^2}{2s}}.
\ee
The
Rodrigues formula takes the form
\be
\label{sama:abelmarie}
e^{-\frac{x^2}{2s}}\,H_n (x,s)= (-sD)^n\, e^{-\frac{x^2}{2s}}
\ee
while the second
order
differential equation satisfied by $H_n(x,s)$ is
\be
\label{second}
\big(sD^2- x D+ n\big)\,H_n(x,s)=0.
\ee
Furthermore, from the relation (\ref{sasama}) we derive the result
\be
\label{labelsaj}
H_n(x+sD,s)\cdot(1)=x^n,
\ee
and the inverse formula for $H_n(x,s)$
\be
\label{labelsaaf}
x^n=n!\,\sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{s^k }{ (2k)!! } \frac{ H_{n-2k}(x,s) }{(n-2k)! }.
\ee
We then obtain
\be
\label{props}
\sum_{k,\,n \,(even)}
\frac{ 1 }{(n-k)!\,k! }=\sum_{k,\,n \,(odd)}
\frac{ 1 }{(n-k)!\,k! },\quad 0\leq k\leq n,\quad n\geq 0.
\ee
From (\ref{sasama}), it is also straighforward to note that the polynomials
$H_n(x,s)$ have an alternative expression given by
\be
\label{mirer}
H_n(x,s)=\exp\left( -s\frac{D^2}{2}\right)\cdot (x^n).
\ee
For any integer $k=0, 1, ..., \lfloor\,n/2\,\rfloor,$ we have the following result
\be
\label{e7j}
D^{2k}\,H_n(x,s)=\frac{n!}{(n-2k)!} H_{n-2k}(x,s).
\ee
\begin{corollary}
The Hermite polynomials $H_n(x,s)$ obey
\be
\label{nversion}
\mathcal{T}_n(s, D)\,H_n(x,s)=x^n
\ee
where the polynomial
\be
\label{samapolynom}
\mathcal{T}_n(\alpha,\beta)= \sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{ 1 }{ (2k)!! } \alpha^k\beta^{2k}.
\ee
\end{corollary}
We are now in a position to formulate and prove the following.
\begin{lemma}
\label{malem}
\be
\mathcal{T}_{2n}(\alpha,\beta)=\frac{(\alpha\beta^2)^n}{(2n)!!}\,{}_2F_0\Bigg(
\begin{array}{c}-n,1\\
-\end{array}\Big|-\frac{2}{\alpha\beta^2}\Bigg)
\ee
and
\be
\mathcal{T}_\infty(\alpha,\beta)=e^{\frac{\alpha\beta^2}{2}}.
\ee
\end{lemma}
{\bf Proof.} From (\ref{samapolynom}), we have
\bea
\label{calcal}
\mathcal{T}_{2n}(\alpha,\beta)&=&\sum_{k=0}^{n}\frac{1}{(2k)!!}(\alpha\beta^2)^{k}\cr
&=&\frac{(\alpha\beta^2)^n}{(2n)!!}\sum_{k=n}^{\infty}\frac{(2n)!!}{(2k)!!}(\alpha\beta^2)^{k-n}.
\eea
By substituting $m=n-k$ in the latter expression and using various identities, we arrive at
\be
\label{calcale}
\mathcal{T}_{2n}(\alpha,\beta)
=\frac{(\alpha\beta^2)^n}{(2n)!!}\sum_{m=0}^{\infty}(-n)_m\;\left(\frac{-2}{\alpha\beta^2}\right)^{m},
\ee
where $(a)_j:=a(a+1)\cdots(a+j-1),\; j \geq 1$ and $(a)_0:=1.$
When $n$ goes to $\infty,$ the polynomial (\ref{samapolynom}) takes the form
\be
\mathcal{T}_\infty(\alpha,\beta)= \sum_{k=0}^{\infty}
\frac{\alpha^k\beta^{2k} }{ (2k)!! } = \sum_{k=0}^{\infty}
\frac{ 1 }{ k!}\left(\frac{\alpha\beta^2}{2}\right)^k
\ee
where $(2k)!!=2^k\, k!$ is used. $\square$
To end this section, let us investigate the Fourier transform of the function $e^{-x^2/2s}H_n(x,s)$. In
\cite{Johann Cigler}, Cigler has proven that
\be
\label{samafouri}
\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}e^{i x y-\frac{x^2}{2s}} d x=
e^{-s\frac{y^2}{2}}.
\ee
Hence,
\be
\label{samaM}
\frac{1}{\sqrt{2\pi\, s}}\int_{\mathbb{R}}e^{i x y+ i(n- 2k)
\kappa x- \frac{x^2}{2s}} d x
=e^{-s\frac{y^2}{2}-(n- 2k)s\,y\,\kappa},
\ee
where $e^{-2s\kappa^2}=1$.
By differentiating the relation (\ref{samafouri}) $2n-2k$ times with respect to $y$, one obtains
\be
\frac{1}{\sqrt{2\pi \,s}}\int_{\mathbb{R}}(-1)^{n-k}x^{2n-2k}e^{ix y-\frac{x^2}{2s}} d x=
D^{2n-2k} e^{-s\frac{y^2}{2}}.
\ee
Evaluating the latter expression at $y=0$ and by making use of (\ref{sama:abelmarie}), one gets
\be
\label{tate}
\frac{(-1)^{n-k}}{\sqrt{2\pi \,s}}\int_{\mathbb{R}} x^{2n-2k}e^{-\frac{x^2}{2s}} d x=
{D^{2n-2k} e^{-s\frac{y^2}{2}}}_{\big|y=0}={(-s)^{2n-2k}H_{2n-2k}(y,s^{-1})e^{-s\frac{y^2}{2}}}_{\big|y=0}.
\ee
\\
\begin{theorem}
\label{sama:propo1}
The Fourier transform of the function $e^{-x^2/2 s}H_{n}(x,s )$ is given by
\be
\label{samafourier}
\frac{1 }{\sqrt{2\pi \,s}}\int_{\mathbb{R}}H_{n} (a\, e^{i \kappa x},s)e
^{ix y-\frac{x^2}{2s}}d x= H_{n}
(a\, e^{-s\,\kappa\, y}, s ) e^{-s\frac{y^2}{2}}
\ee
where $a$ is an arbitrary constant factor. For $y=0,$ we have
\be
\label{sama:fourier}
\frac{1 }{\sqrt{2\pi\, s}}\int_{\mathbb{R}}H_{n} (x,s)e
^{-\frac{x^2}{2s}}d x= 0.
\ee
\end{theorem}
{\bf Proof.}
Using (\ref{sasama}) and (\ref{samaM}), we obtain
\bea
\frac{1}{\sqrt{2\pi\, s}}\int_{\mathbb{R}}H_{n}(a \,e^{i\kappa x},s)
e^{ix y-\frac{x^2}{2\,s}}d x &=&\sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^{k}n!\,s^k\,a^{n-2k}}{(n-2k)!\,(2k)!!}
\frac{1}{\sqrt{2\pi\, s}}\int_{\mathbb{R}}e^{i x y+ i(n- 2k)
\kappa x- \frac{x^2}{2s}} d x \cr
& =& \sum_{k=0}^{\lfloor
n/2\,\rfloor}\frac{(-1)^{k}\,n!\;s^k\,a^{n-2k}}{(n-2k)!\,(2k)!!}
e^{-\frac{s}{2}[\kappa (n-2k)+y]^2} \cr
& =&e^{-s\frac{y^2}{2}} H_{n} (a\, e^{-s\,\kappa\, y}
, s ).
\eea
Combining (\ref{sasama}) and (\ref{tate}) for $n=2n$, we have
\bea
\frac{1}{\sqrt{2\pi \,s}}\int_{\mathbb{R}}H_{2n}(x,s)
e^{-\frac{x^2}{2\,s}}d x &=&\sum_{k=0}^n
\frac{(-1)^k\,(2n)!\,s^k}{(2n-2k)!\,(2k)!!}\,
\frac{1}{\sqrt{2\pi\, s}}\int_{\mathbb{R}}x^{2n-2k}\,e^{ixy- \frac{x^2}{2\,s}} d x_{\big|y=0} \cr
& =&(-1)^n\sum_{k=0}^n
\frac{ (2n)!\,s^k}{(2n-2k)!\,(2k)!!}{ D^{2n-2k} e^{-s\frac{y^2}{2}}}_{\big|y=0} \cr
& =&(-1)^{n}\,s^{2n}\,e^{-s\frac{y^2}{2}}\sum_{k=0}^n
\frac{(2n)!\,s^{-k}}{(2n-2k)!\,(2k)!!}{H_{2n-2k}(y,s^{-1})}_{\big|y=0}\cr
& =&s^{2n}\,(2n)! \,\sum_{k=0}^n
\frac{(-1)^k}{(2n-2k)!!\,(2k)!!} \cr
&=&0
\eea
where (\ref{props}) is used. $\square$
\section{New $q-$Hermite polynomials $H_n(x,s|q)$}
\label{sction2}
In this section, we construct
through the $q-$chain rule
a new family of $q-$Hermite polynomials denoted by
$H_n(x,s|q).$
We first introduce some standard $q-$notations.
For $n\geq 1,\;q\in\mathbb{C}$,
we denote the $q-$deformed number \cite{Ernst} by
\be
\{n\}_{q}:=\sum_{k=0}^{n-1}q^k.
\ee
In the same way, we define the $q-$factorials
\be
\{n\}_{q}!:=\prod_{k=1}^n \{k\}_{q}, \quad \{2n\}_{q}!!:=\prod_{k=1}^n \{2k\}_{q},\quad \{2n-1\}_{q}!!:=\prod_{k=1}^n \{2k-1\}_{q}
\ee
and, by convention,
\be
\{0\}_{q}!:=1=: \{0\}_{q}!! \quad \mbox{ and } \quad \{-1\}_{q}!!=1.
\ee
For any positive number $c,$ the $q-$Pochhammer symbol $\{c\}_{n,q}$
is defined as follows:
\be
\{c\}_{n,q}:=\prod_{k=0}^{n-1} \{c+k\}_{q}
\ee
while the $q-$binomial coefficients are defined by
\bea
\label{samasama:sa}
{n\atopwithdelims\{\} k}_q:=\frac{ \{n\}_{q}!}{ \{n-k\}_{q}! \{k\}_{q}!}=\frac{(q;q)_n}{(q;q)_{n-k}(q;q)_k}\quad \mbox{ for }\quad
0\leq k\leq n,
\eea
and zero otherwise, where $(a;q)_n:=\prod_{k=0}^{n-1}(1-a\,q^k),\;(a;q)_0:=1.$
\begin{definition}\cite{HahnW, sama}
The Hahn $q-$addition $\op$ is the function: $\mathbb{C}^3\rightarrow \mathbb{C}^2$
given by:
\be
(x,y,q)\mapsto (x,y)\equiv x\op y,
\ee
where
\bea
\label{addition}
(x \op y)^n:&=&(x+y)(x+q \,y)\ldots (x+q^{n-1}\,y)\cr
&=&\sum_{k=0}^n{n
\atopwithdelims\{\} k}_q\, q^{({}^k_2)}\,x^{n-k}\,y^k,\quad n\geq 1,\quad (x\op y)^0:=1,
\eea
while the $q-$subtraction $\om$ is defined as follows:
\be
\label{additionm}
x \om y:= x\op(-y).
\ee
\end{definition}
Consider a function $F$
\bea
F:D_R \longrightarrow\mathbb{C},\quad z \longmapsto \sum_{n=0}^\infty c_n\, z^n,
\eea
where $D_R$ is a disc of radius $R.$ We define $F(x\op y)$ to mean the formal series
\bea
\sum_{n=0}^\infty c_n(x\op y)^n\equiv\sum_{n=0}^\infty \sum_{k=0}^n\,c_n\,{n
\atopwithdelims\{\} k}_q\, q^{({}^k_2)}\,x^{n-k}\,y^k.
\eea
Let $e_q,\;E_q,\; \cos_q$ and $\sin_q$ be the fonctions defined as follows:
\bea
e_q (x) :&=& \sum_{n=0}^{\infty} \frac{1}{ \{n\}_{q}!}x^n\\
E_q (x) :&=& \sum_{n=0}^{\infty} \frac{q^{n(n-1)/2}}{ \{n\}_{q}!}x^n\cr
\cos_q(x):&=&\frac{e_q (i\,x)+e_q (-i\,x)}{2}= \sum_{n=0}^{\infty} \frac{(-1)^n}{ \{2n\}_{q}!}x^{2n}\\
\sin_q(x):&=&\frac{e_q (i\,x)-e_q (-i\,x)}{2\,i}= \sum_{n=0}^{\infty} \frac{(-1)^n}{ \{2n+1\}_{q}!}x^{2n+1}.
\eea
We immediately obtain the following rules for the product of two exponential functions
\be
e_q (x) E_q (y) = e_q ( x \op y ).
\ee
The new family of $q-$Hermite polynomials $H_n(x,s|q) $ can be determined
by the generating function
\be
\label{functiongeneratrice}
e_q\big( tx \ominus_{q,q^2} st^2/ \{2\}_{q} \big) =
e_q( tx)E_{q^2} (- st^2/ \{2\}_{q} ) := \sum_{n=0}^{\infty} \frac{H_n(x,s|q) }{ \{n\}_{q}!}t^n,\quad |t| <1,
\ee
where \cite{sama}
\be
(a\ominus _{q, q^{2}}b )^n := \sum_{k=0}^{n} \frac{ \{n\}_{q}!}{ \{n-k\}_{q}! \, \{k\}_{q^{2} }!}
(-1)^k q^{k(k-1)} a^{n-k} \,b^{k},\quad (a\ominus_{q, q^{2}} b )^0 := 1.
\ee
Performing the $q-$derivative $D_x^q$ of both sides
of (\ref{functiongeneratrice}) with respect to $x$, one obtains
\bea
\label{lowering}
D_x^q\, H_n(x,s|q)= \{n\}_{q}\, H_{n-1}(x,s|q),
\eea
where
\be
\label{lowm}
D_x^q\, f(x)=\frac{f(x)-f(qx)}{(1-q)x}
\ee
satisfying
\bea
D_x^q (a\, x \op b)^n= \{n\}_{q}\,(a\, x \op b)^{n-1}.
\eea
Recall \cite{Ismails} that the Al-Salam-Chihara polynomials $P_n(x;a,b,c)$ satisfy the following recursion relation:
\be
\label{sama:al}
P_{n+1}(x;a,b,c)=(x-a\,q^n)\,P_n(x;a,b,c)-(c+b\,q^{n-1})\, \{n\}_{q}\,P_{n-1}(x;a,b,c)
\ee
with $P_{-1}(x;a,b,c)=0$ and $P_0(x;a,b,c)=1$.\\
Performing the $q-$derivative
of both sides of
(\ref{functiongeneratrice}) with respect to $t$, we have
\bea
\label{raising}
H_{n+1}(x,s|q)= x\, H_{n}(x,s|q)-s\, \{n\}_{q} \,q^{n-1}\,H_{n-1}(x,s|q),\quad n\geq 1
\eea
with $H_0(x,s|q):=1.$ \\
By setting
$a=0=c$ and $b=s$ in (\ref{sama:al}), one obtains the recursion relation (\ref{raising}).
From the latter equation,
one can see that
\be
\label{sama:qinitial}
H_{2n}(0,s|q)=(-s)^n\,q^{n(n-1)}\, \{2n-1\}_{q}!!,\quad H_{2n+1}(0,s|q)=0.
\ee
The first fourth new polynomials are given by
\bea
H_1(x,s|q) &=& x,\\
H_2(x,s|q)& =& x^2 -s,\\
H_3(x,s|q)&=& x^3 - \{3\}_{q}s x,\\
H_4(x,s|q)&=& x^4 -(1+q^2) \{3\}_{q}sx^2+ q^2\,\{3\}_{q}s^2.
\eea
More generally, we have the following.
\begin{theorem}
The explicit formula for the new Hermite polynomials $H_n(x,s|q)$ is given by
\bea
\label{sama:qhermite}
H_n(x,s|q) &=& \sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{ (-1)^k q^{k(k-1)} \{n\}_{q}! }{ \{n-2k\}_{q}! \, \{2k\}_{q}!! }s^k x^{n-2k}\\
&=&x^n\,{}_2\phi_0\Bigg(\begin{array}{c}q^{-n},q^{1-n}\\
-\end{array}\Big|\;q^2;\;\frac{sq^{2n-1}}{(1-q)x^2}\Bigg),
\eea
where ${}_2\phi_0$ is the $q-$hypergeometric series \cite{ASK}.
\end{theorem}
{\bf Proof.} Expanding the generation function
given in (\ref{functiongeneratrice}) in Maclaurin series, we have
\bea
\label{sammas}
e_q( t\,x)E_{q^2} (- s\,t^2 /\{2\}_{q}) &=& \sum_{k=0}^{\infty} \frac{( x\,t)^k }{\{k\}_{q}!}
\sum_{m=0}^{\infty} \frac{(-1)^m q^{m(m-1)} }{\{m\}_{q^2} ! } \left(\frac{s\,t^2}{\{2\}_{q}}\right)^m \cr
&=& \sum_{k=0}^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m q^{m(m-1)}
x^k }{\{k\}_{q}! \,\{m\}_{q^2} ! } \left(\frac{s}{\{2\}_{q}}\right)^m t^{k+ 2m}.
\eea
By substituting
\be
k +2m =n ~\Rightarrow~ m \le\lfloor\,n/2\,\rfloor,
\ee
and
\be
\{2\}_{q}\,\{m\}_{q^2}=\{2m\}_{q}
\ee
in (\ref{sammas}), we have
\be
e_q( t\,x)E_{q^2} (- s\,t^2/\{2\}_{q} ) = \sum_{n=0}^{\infty} \lb
\sum_{m=0}^{\lfloor\,n/2\,\rfloor} \frac{(-1)^m q^{m(m-1)}
s^m\,x^{n-2m} }{\{n-2m\}_{q}!\, \{2m\}_{q} !! }\rb t^{n},
\ee
which achieves the proof. $\square$
In the limit case when $x\to \{2\}_{q}\,x,\;s\to (1-q)\,\{2\}_{q},$ the
polynomials $H_n(x,s|q)$ are reduced to $H_n^q(x)$ investigated by Chung et {\it al} \cite{sama}. When $s\to 1-q$, they are
reduced to the discrete $q-$Hermite I polynomials \cite{ASK}.\\
The relation (\ref{raising}) allows us to write
\be
H_{n}(x,s|q)=(x-sq^N\circ D_x^q)\,H_{n-1}(x,s|q),
\ee
where
the operator $N$ acts on the polynomials $H_n(x,s|q)$ as follows:
\be
NH_n(x,s|q):= n\,H_n(x,s|q),\quad q^N\circ D_x^q=D_x^q\circ q^{N-1}.
\ee
It is straightforward to show that the polynomials (\ref{sama:qhermite}) satisfy the
following $q-$difference equation
\be
\label{sama:hounk}
\big(s\, (D_x^q)^2 -x\,q^{2-n}\,D_x^q+q^{2-n}\,\{n\}_{q}\big)\,H_n(x,s|q)=0.
\ee
In the limit case when $q$ goes to $1$, the $q-$difference
equation (\ref{sama:hounk}) reduces to the well-known differential equation (\ref{second}).
For $n$ even or odd, the polynomials $H_{n}(x,s|q)$ obey the following generating functions
\be
\sum_{n=0}^\infty\frac{H_{2n}(x,s|q)}{\{2n\}_{q}!}(-t)^n=\cos_q(x\sqrt{t})\,E_{q^2}(s\,t/\{2\}_{q}),\; |t| <1
\ee
or
\be
\sum_{n=0}^\infty\frac{H_{2n+1}(x,s|q)}{\{2n+1\}_{q}!}(-t)^n=
\frac{1}{\sqrt{t}}\sin_q(x\sqrt{t})\,E_{q^2}(s\,t/\{2\}_{q}), \; |t| <1,
\ee
respectively.
\begin{theorem}
\label{theo:poly}
The polynomials $H_n(x,s|q)$ can be expressed as
\be
H_n(x,s|q) =\prod_{k=1}^{n} \big( x -s\,q^{n-1-k}\, D_x^q \big)\cdot(1).
\ee
We also have
\bea
\label{sama:reso}
H_n \left(x+sq^N\circ D_x^q,s|q\right)\cdot(1)=x^n.
\eea
\end{theorem}
{\bf Proof.}
Since (\ref{lowering}) and (\ref{raising}) are satisfied, we have
\bea
H_{n}(x,s|q)&=& x\ H_{n-1}(x,s|q)-s\, q^{n-2}\, \{n-1\}_{q}\, H_{n-2}(x,s|q)\cr
&=& x \,H_{n-1}(x,s|q)-s q^{n-2}\,D_x^q\,H_{n-1}(x,s|q).
\eea
The rest holds by induction on $n$.\\
To prove the relation (\ref{sama:reso}) we replace $x^{n-2k}$ in (\ref{sama:qhermite}) by
$(x+sq^N\circ D_x^q)^{n-2k}$ and apply the corresponding linear operator
to $1$. The relation (\ref{sama:reso}) is true for $n=0$ and $n=1.$
For $n=2$, we have
\bea
H_2\left(x+sq^N\circ D_x^q,s|q\right)\cdot(1)&=&\left(x+sq^N\circ D_x^q\right)^2\cdot(1)-s\cr
&=&\left(x+sq^N\circ D_x^q\right)\cdot (x)-s\cr
&=&x^2.
\eea
Assume
that (\ref{sama:reso}) is true for $n-1,\;n\geq 3.$
Then we must prove that
\be
H_{n} \left(x+sq^N\circ D_x^q,s|q\right) \cdot(1)=x^{n}.
\ee
From
(\ref{raising}), we have
\bea
H_{n} \left(x+sq^N\circ D_x^q,s|q\right) \cdot1&=& \left(x+sq^N\circ D_x^q\right)H_{n-1}
\left(x+sq^N\circ D_x^q,s|q\right) \cdot(1)\cr
&-&s\{n-1\}_{q}\,q^{n-2}H_{n-2} \left(x+sq^N\circ D_x^q,s|q\right)\cdot(1)\cr
&=&\left(x+sq^N\circ D_x^q\right)\cdot x^{n-1}
-s\{n-1\}_{q}\,q^{n-2} x^{n-2}\cr
&=&x^{n}
\eea
which achieves the proof. $\square$\\
From the {\bf Theorem} \ref{theo:poly}, we obtain the following.
\begin{corollary}
\label{sama:theorem}
The polynomials (\ref{sama:qhermite}) have the following inversion formula
\be
\label{inversion}
x^n = \{n\}_{q}!\sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{ q^{k(k-1)} \;s^k }{ \{2k\}_{q}!! }\frac{ H_{n-2k}(x,s|q) }{\{n-2k\}_{q}! }.
\ee
\end{corollary}
{\bf Proof.} Let $h_n^q(x,s)$ be the polynomial defined by
\be
h_n^q(x,s)=\left(x+sq^N\circ D_x^q\right)^n\cdot(1).
\ee
Note that
$
h_n^q(x,-s)=H_n(x,s|q).
$
From (\ref{sama:reso}), we have
\bea
x^n&=&\sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{(-1)^k q^{k(k-1)}\{n\}_{q}! }{\{n-2k\}_{q}! \{2k\}_{q}!! }s^k
\left(x+sq^N\circ D_x^q\right)^{n-2k}\cdot(1)\cr
&=&\sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{ q^{k(k-1)}\{n\}_{q}! \,s^k }{\{n-2k\}_{q}!\{2k\}_{q}!! } h_{n-2k}^q(x,-s)
\eea
which achives the proof.
$\square$
From (\ref{lowering}), one readily deduces that, for integer
powers $k=0, 1, ..., \lfloor\,n/2\,\rfloor$ of the operator
$D_x^q,$
\be
\label{e7}
(D_x^q)^{2k}H_n(x,s|q)=\gamma_{n,k}(q)H_{n-2k}(x,s|q),\quad
\gamma_{n,k}(q)=\frac{\{n\}_{q}!}{\{n-2k\}_{q}!}.
\ee
Therefore, we have the following decomposition of unity
\be
\sum_{k=0}^{\lfloor\,n/2\,\rfloor}\frac{ (-1)^kq^{k(k-1)} s^k }{ \{2k\}_{q}!!
}(D_x^q)^{2k}\sum_{m=0}^{\lfloor\,n/2\rfloor}
\frac{ q^{m(m-1)} s^m }{ \{2m\}_{q}!! }(D_x^q)^{2m}={\bf 1}
\ee
and the new
$q-$Hermite polynomials $H_n(x,s|q)$
obey
\bea
\label{rinversion}
\mathcal{L}_n(s, D_x^q|q)H_n(x,s|q)=x^n
\eea
where the polynomial $\mathcal{L}_n(\alpha,\beta|q)$ is defined as follows:
\be
\label{sama:polynom}
\mathcal{L}_n(\alpha,\beta|q)= \sum_{k=0}^{\lfloor\,n/2\,\rfloor}
\frac{ q^{k(k-1)} }{ \{2k\}_{q}!! } \alpha^k\beta^{2k}.
\ee
This polynomial is essentially the $(q,n)-$exponential
function $E_{q,n}(x)$ investigated by Ernst \cite{Ernst}, i.e.,
$\mathcal{L}_{n-1}(\alpha,\beta|q)=E_{q^{-2},\lfloor\,n/2\,\rfloor}(\alpha\beta^2/\{2\}_{q}).$
We are now in a position to formulate and prove the following.
\begin{lemma}
From the polynomial (\ref{sama:polynom}) we have
\be
\mathcal{L}_{2n}(\alpha,\beta|q)=\frac{(\alpha\beta^2)^nq^{n(n-1)}}{\{2n\}_{q}!!}{}_3\phi_2\Bigg(
\begin{array}{c}q^{-n},-q^{-n},q\\
0,0\end{array}\Big|q;-\frac{q^2}{(1-q)\alpha\beta^2}\Bigg)
\ee
and
\be
\mathcal{L}_\infty(\alpha,\beta|q)=E_{q^2}(\alpha\beta^2/\{2\}_{q}).
\ee
\end{lemma}
{\bf Proof.} As it is defined in (\ref{sama:polynom}), we have
\begin{eqnarray}
\mathcal{L}_{2n}(\alpha,\beta|q)&=&\sum_{k=0}^{n}\frac{q^{ k(k-1)}}{\{2k\}_{q}!!}(\alpha\beta^2)^{k}\cr
&=&\frac{(\alpha\beta^2)^n}{\{2n\}_{q}!!}\sum_{k=n}^{\infty}\frac{q^{ k(k-1)}\{2n\}_{q}!!}{\{2k\}_{q}!!}(\alpha\beta^2)^{k-n}.
\end{eqnarray}
By substituting
$m=n-k$ in the latter expression, we arrive at
\begin{eqnarray}
\mathcal{L}_{2n}(\alpha,\beta|q)
&=&\frac{(\alpha\beta^2)^n}{\{2n\}_{q}!!}\sum_{m=0}^{\infty}
\frac{q^{ (n -m)(n -m-1)}\{2n\}_{q}!!}{\{2n-2m\}_{q}!!}(\alpha\beta^2)^{-m}\cr
&=&\frac{(\alpha\beta^2)^nq^{n(n-1)}}{\{2n\}_{q}!!}\sum_{m=0}^{\infty}
(q^{-2n};q^2)_m\left(-\frac{q^2}{(1-q)\alpha\beta^2}\right)^m.
\end{eqnarray}
When $n\to\infty$, (\ref{sama:polynom}) takes the form
\be
\mathcal{L}_\infty(\alpha,\beta|q)= \sum_{k=0}^{\infty}
\frac{ q^{k(k-1)} }{ \{2n\}_{q}!! } (\alpha\beta^2)^k = \sum_{k=0}^{\infty}
\frac{ q^{k(k-1)} }{ \{k\}_{q^2 }!}\left(\frac{\alpha\beta^2}{\{2\}_{q}}\right)^k
\ee
which achieves the proof. $\square$
In the limit, when $q\to1$, the polynomial $\mathcal{L}_n(\alpha,\beta|q)$ is reduced
to the classical one's $\mathcal{T}_n(\alpha,\beta)$, i.e., $\lim_{q\to 1}
\mathcal{L}_n(\alpha,\beta|q)=\mathcal{T}_n(\alpha,\beta),\;\forall\,n.$
\section{Fourier transforms of the new $q-$Hermite
polynomials $H_n(x,s|q)$ }
\label{sction3}
In this section,
we compute the Fourier integral
transforms associated to the new $q-$Hermite
polynomials $H_n(x,s|q)$.
\subsection{ $q^{-1}-$Hermite polynomials $H_n(x,s|q^{-1})$ }
Let us rewrite the new $q-$Hermite polynomials
(\ref{sama:qhermite}) in the following form
\be
\label{sama:pipp}
H_{n}(x,s|q)= \sum_{k=0}^{\lfloor\,n/2\,\rfloor}
c_{n,k}(q)\,s^{k}x^{n-2k},
\ee
where the associated coefficients
$c_{n,\,k}(q)$ are given by
\be
\label{sama:coef}
c_{n,\,k}(q):= \frac{ (-1)^k q^{k(k-1)}\,\{n\}_{q}!
}{\{n-2k\}_{q}! \,\{2k\}_{q}!! }.
\ee
By a direct computation, one can easily check that these coefficients
satisfy the following recursion relation
\be
\label{rett}
c_{n+1,\,k}(q)=c_{n,\,k}(q)-q^{n-1}\{n\}_{q}\,c_{n-1,\,k-1}(q),
\ee
with $
c_{0,\,k}(q)=\delta_{0,k}, \;\; c_{n,\,0}(q)=1.$\\
From the definition of the $q-$binomial coefficients
in (\ref{samasama:sa}), it is not hard to derive an
inversion formula
\bea
\label{samasamasa}
{n\atopwithdelims\{\} 2k}_{q^{-1}}=
q^{2k(2k-n)}\,{n\atopwithdelims\{\} 2k}_q,\qquad
0\leq k\leq \lfloor n/2\rfloor.
\eea
Then, one readily deduces that
\be
\label{sama:labas}
c_{n,\,k}(q^{-1})= q^{k(k+3-2n)}c_{n,\,k}(q),
\ee
allowing to define the $q^{-1}-$Hermite
polynomials $H_n(x,s|q^{-1})$ in the following form
\be
\label{sama:inm}
H_n(x,s|q^{-1}):= \sum_{k=0}^{\lfloor\,n/2\,\rfloor}\,c_{n,\,k}(q^{-1})\,s^{k}x^{n-2k}. \ee
The recursion relation
\be
\label{sama:lab}
c_{n+1,\,k}(q^{-1})= q^{-2k}\,c_{n,\,k}(q^{-1})- q^{3-n-2k}\,\{n\}_{q}\,c_{n-1,\,k-1}(q^{-1}),\quad n\geq 1
\ee
is valid for the coefficients (\ref{sama:labas})
with $c_{0,\,k}(q^{-1})=q^{k(k+3)}\,\delta_{0,k}, \;\; c_{n,\,0}(q^{-1})=1.$ \\
Since (\ref{sama:lab}) is satisfied, the $q^{-1}-$Hermite polynomials $H_n(x,s|q^{-1} )$ obey
the relation
\be
\label{samanm}
H_{n+1}(x,s|q^{-1})=xH_n(x,sq^{-2}|q^{-1})-sq^{1-n}\{n\}_{q}\,H_{n-1}(x,sq^{-2}|q^{-1}),\quad n\geq 1,
\ee
with $H_0(x,sq^{-2}|q^{-1}):=1.$\\
The action of the operator $D_x^q$ on
the polynomials (\ref{sama:inm}) is given by
\bea
\label{samanma}
D_x^q\,H_n(x,s|q^{-1})=\{n\}_{q}\,H_{n-1}(x,sq^{-2}|q^{-1}).
\eea
Let $\epsilon$
denote the operator which maps $f(s)$ to $f(qs)$. Then, from (\ref{samanm}) and (\ref{samanma}) one
can establish that
\be
H_{n}(x,s|q^{-1}) =\prod_{k=1}^{n} \big(x\,\epsilon^{-2}-sq^{k+1-n}D_x^q\big)\cdot(1).
\ee
\subsection{Fourier transforms of the new $q-$Hermite polynomials $H_n(x,s|q)$ }
Considering
the well-known Fourier transforms (\ref{samafouri})
for the Gauss exponential function $e^{-x^2/2s},$ the
Fourier integral transforms
for the exponential function $\exp(i(n-2k)\kappa x-x^2/2s)$ is computed as follows:
\be
\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}e^{i x y+ i(n- 2k)
\kappa x- \frac{x^2}{2s}} d x
=q^{\frac{n^2}{4}+k(k-n)}e^{-s\frac{y^2}{2}-(n- 2k)sy\kappa},
\ee
where $q=e^{-2s\kappa^2} \leq1$ and $0\leq \kappa < \infty$.
\begin{theorem}
\label{sama:propo1}
The new $q-$Hermite polynomials $H_n(x,s|q)$ and
$H_n(x,s|q^{-1})$ defined in (\ref{sama:pipp}) and (\ref{sama:inm}),
respectively, are connected by the integral Fourier transform of the
following form
\be
\label{sama:fourier}
\frac{1 }{\sqrt{2\pi s}}\int_{\mathbb{R}}H_{n} (b e^{i \kappa x},s|q)e
^{ix y-\frac{x^2}{2s}}d x= q^{\frac{n^2}{4}}H_{n}
(b e^{-s\kappa y}, q^{n-3} s| q^{-1})\, e^{-s\frac{y^2}{2}}
\ee
where $b$ is an arbitrary constant factor.
\end{theorem}
{\bf Proof.} To prove this theorem, let us make use of (\ref{sama:pipp}) and
evaluate the left hand side of (\ref{sama:fourier}). Then,
\bea
\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}H_{n}(b e^{i\kappa x},s|q)
e^{ix y-\frac{x^2}{2s}}d x &=&\sum_{k=0}^{\lfloor n/2\rfloor}c_{n,k} (q)s^{k}
b^{n-2k}\,\frac{1}{\sqrt{2\pi s}}\int_{\mathbb{R}}e^{i x y+ i(n- 2k)
\kappa x- \frac{x^2}{2s}} d x \cr
& =&\sum_{k=0}^{\lfloor
n/2\,\rfloor}c_{n,k}(q)s^k b^{n-2k}e^{-\frac{s}{2}[\kappa (n-2k)+y]^2} \cr
& = &q^{\frac{n^2}{4}}\sum_{k=0}^{\lfloor
n/2\,\rfloor}c_{n,k}(q)q^{-k(n-k)}s^k b^{n-2k}e^{-s\frac{y^2}{2}-(n- 2k)sy\kappa} \cr
& = &q^{\frac{n^2}{4}}\sum_{k=0}^{\lfloor
n/2\,\rfloor}c_{n,k}(q^{-1})(q^{n-3}s)^k (be^{-sy\kappa})^{n-2k} e^{-s\frac{y^2}{2}}\cr
& =& q^{\frac{n^2}{4}}H_{n}
(b e^{-s\kappa y}, q^{n-3} s |q^{-1})\, e^{-s\frac{y^2}{2}}.
\eea
$\square$
\section{Doubly indexed Hermite polynomials $\mathcal{H}_{n,p}(x,s|q)$}
\label{sction4}
In this section, we construct a novel family of Hermite polynomials
called {\it doubly indexed Hermite polynomials}, $\mathcal{H}_{n,p}(x,s|q).$
First, let us
defined the
$(q;p)-$shifted factorials $ (a;q)_{pk}$
and the $(q;p)-$number as follows:
\be
\label{sama:newdef}
(a;q)_0:=1,\quad (a;q)_{pk}:=(a,aq,\cdots,aq^{p-1};q^p)_k, \quad p\geq 1,\;k=1,2,3,\cdots
\ee
and
\be
\{pk\}_{q}:=\frac{1-q^{pk}}{1-q},\quad \{pk\}_{q}!!:=\prod_{l=1}^k \{pl\}_{q},\quad \{0\}_{q}!!:=1,
\ee
respectively.
\begin{definition}
For a positive integer $p$, a class of doubly indexed
Hermite polynomials $\big\{\mathcal{H}_{n,p}\big\}_{n,p}$ is defined
such that
\be
\label{sama:doublyH}
\mathcal{H}_{n,p}(x,s|q) := E_{q^p} \lb - s\frac{(D_x^q )^p}{ \{p\}_{q} } \rb\cdot (x ^n).
\ee
\end{definition}
If $p=2,$ a subclass of the polynomials
(\ref{sama:doublyH}) is reduced to the class of polynomials
(\ref{sama:qhermite}).
More generally,
their explicit formula is given by
\bea
\label{sama:qdoublyH}
\mathcal{H}_{n,p}(x,s|q)&=& \{n\}_{q}!\sum_{k=0}^{\lfloor\,n/p\,\rfloor}
\frac{ (-1)^k q^{p({}^k_2)} \,s^k}{ \{pk\}_{q}!!}\frac{ x^{n-pk} }{ \{n-pk\}_{q}! }\\
&=&x^n{}_p\phi_0\left(\begin{array}{c}q^{-n},q^{-n+1},\cdots,q^{-n+p-1}\\
-\end{array}\Big|\;q^p;\;\frac{sq^{p(n+(1-p)/2)}}{(1-q)^{p-1}x^p}\right),
\eea
where ${}_p\phi_0$ is the $q-$hypergeometric series \cite{ASK}.
Since $D_x^qe_q(\omega\,x)=\omega\,e_q(\omega\,x),$ we derive
the generating function of the
polynomials (\ref{sama:doublyH})
as
\be
\label{sama:functionqdoublygeneratrice}
f_q(x,s;p):=e_q( tx)E_{q^p} (- st^p/ \{p\}_{q} ) =
\sum_{n=0}^{\infty} \frac{\mathcal{H}_{n,p}(x,s|q) }{ \{n\}_{q}!}t^n, \quad |t| < 1.
\ee
These polynomials are the solutions of the $q-$analogue of the generalized heat equation \cite{Dattoli}
\be
(D_x^q)^p f_q(x,s;p)= -\{p\}_qD_s^qf_q(x,s;p),\quad f_q(x,0;p)=x^n.
\ee
For any real number $c$ and a positive integer $p$, $|q| <1,$ we have
\be
\sum_{n=0}^{\infty} \frac{\{c\}_{n,q}\mathcal{H}_{n,p}(x,s|q) }{\{n\}_{q}!}t^n=
\frac{1}{(xt;q)_c}{}_p\phi_p\Bigg(\begin{array}{c}q^{c},q^{c+1},\cdots,q^{c+p-1}\\
xtq^{c},xtq^{c+1},\cdots,xtq^{c+p-1}\end{array}\Big|\;q^p;\; \frac{s\,t^p}{(1-q)^{p-1}}\Bigg), \;|xt| <1.
\ee
Performing the $q-$derivative of both sides
of (\ref{sama:functionqdoublygeneratrice}) with respect to $x$ and $t$, one obtains
\be
\label{sa:lowering}
D_x^q \,\mathcal{H}_{n,p}(x,s|q)= \{n\}_{q} \,\mathcal{H}_{n-1,p}(x,s|q)
\ee
and
\be
\label{sa:raising}
\mathcal{H}_{n+1,p}(x,s|q)= x \mathcal{H}_{n,p}(x,s|q)-s q^{n-p+1}
\{n\}_{q} \{n-1\}_{q} \cdots \{n-p+2\}_{q} \mathcal{H}_{n-p+1}(x,s|q),\; n\geq 1,
\ee
with $\mathcal{H}_{0,p}(x,s|q):=1.$
The polynomials (\ref{sama:doublyH}) obey the following $p-$th order difference equation
\be
\Big( s\,(D_x^q)^p-q^{p-n}\,x\,D_x^q+q^{p-n}\, \{n\}_{q}\Big)\,\mathcal{H}_{n,p}(x,s|q)=0.
\ee
\section{Concluding remarks}
In this paper, we have constructed a family of new $q-$Hermite polynomials
$H_n(x,s|q)$.
Several properties related to these
polynomials have been computed and discussed.
Finally, we have constructed
a novel family of Hermite polynomials $\mathcal{H}_{n,p}(x,s|q)$
called doubly indexed Hermite polynomials.
In the limit cases, when $q$ goes to $1$ and $s$ goes to $-py,$ the
polynomials $\mathcal{H}_{n,p}(x,s|q)$ are reduced to
the higher-order Hermite polynomials, sometimes
called the Kamp\'e de F\'eriet or the Gould Hopper polynomials \cite{Dattoli}-\cite{GOULDHOPPER}, i.e.,
\be
\mathcal{H}_{n,p}(x,-py|1)\equiv g_{n}^p(x,y):= n!\sum_{k=0}^{\lfloor\,n/p\,\rfloor}
\frac{ y^kx^{n-pk}}{ k! \, (n-pk)! }.
\ee
When $q$ goes to $1,\;x\to px$ and $s\to p,$ the polynomials $\mathcal{H}_{n,p}(x,s|q)$
become the Hermite
polynomials investigated by Habibullah
and Shakoor \cite{Habibullah}, i.e.,
\be
\mathcal{H}_{n,p}(px,p|1)\equiv S_{p,n}(x):= n!\sum_{k=0}^{\lfloor\,n/p\,\rfloor}
\frac{ (-1)^k(px)^{n-pk}}{ k! \, (n-pk)! }.
\ee
For $p=2$, the doubly indexed polynomials $\mathcal{H}_{n,p}(x,s|q)$ are reduced to the new $q-$Hermite polynomials $H_{n}(x,s|q),$ i.e.,
$\mathcal{H}_{n,2}(x,s|q)\equiv H_{n}(x,s|q)$.
\section*{Acknowledgements}
This work is partially supported by the Abdus Salam International
Centre for Theoretical Physics (ICTP, Trieste, Italy) through the
Office of External Activities (OEA)-\mbox{Prj-15}. The ICMPA
is in partnership with
the Daniel Iagolnitzer Foundation (DIF), France. | 137,553 |
\begin{document}
\hbox{}
{\parindent=0pt
{\large \bf Lagrangian and Hamiltonian dynamics of
submanifolds}
\bigskip
{\bf G.Giachetta}, {\bf L.Mangiarotti}$^1$, {\bf G.Sardanashvily}$^2$
\bigskip
\begin{small}
$^1$ Department of Mathematics and Informatics, University of
Camerino, 62032 Camerino (MC), Italy
\medskip
$^2$ Department of Theoretical Physics, Moscow State University,
117234 Moscow, Russia
\bigskip
{\bf Abstract}
Submanifolds of a manifold are described as sections of a certain fiber
bundle that enables one to consider their Lagrangian and (polysymplectic)
Hamiltonian dynamics as that of a particular classical field theory.
In particular, their Lagrangians and Hamiltonians must satisfy rather
restrictive Noether identities. For
instance, this is the case of relativistic mechanics and classical string
theory.
\end{small}
\bigskip
}
\section{Introduction}
As is well known, fiber bundles and jet manifolds of their sections
provide an adequate mathematical formulation of classical field theory.
In particular, field Lagrangians and their Euler--Lagrange operators are
algebraically described as elements of the variational bicomplex
\cite{tak2,ander,jmp01}. This description is extended to Lagrangian
theory of odd fields \cite{barn,cmp04,jmp05}.
The Hamiltonian counterpart of first-order Lagrangian theory on fiber
bundles is covariant Hamiltonian formalism developed in
multisymplectic, polysymplectic, and Hamilton -- De Donder)
variants (see, e.g., \cite{book,jpa99,krup,ech,leon}).
Jets of sections of fiber bundles are particular jets of submanifolds.
Namely, a space of jets of submanifolds admits a cover by charts of jets
of sections \cite{book,kras,modu}. Three-velocities in relativistic
mechanics exemplify first order jets of submanifolds
\cite{sard98,book98}. A problem is that differential forms on jets of
submanifolds do not constitute a variational bicomplex because horizontal
forms (e.g., Lagrangians) are not preserved under coordinate
transformations.
We consider $n$-dimensional submanifolds of an $m$-dimensional smooth
real manifold $Z$, and associate to them sections of a trivial fiber
bundle $Z_Q=Q\times Z\to Q$, where $Q$ is some $n$-dimensional manifold.
Here, we restrict our consideration to first order jets of submanifolds,
and state their relation to jets of sections of the fiber bundle $Z_Q\to
Q$ (the formulas (\ref{s17}), (\ref{s31}), and Proposition 1).
This
relation fails to be one-to-one correspondence. The ambiguity contains,
e.g., diffeomorphisms of $Q$.
Then Lagrangian and (polysymplectic)
Hamiltonian formalism on a fiber bundle $Z_Q\to Q$ is developed in a
standard way, but Lagrangians and Hamiltonians are required to be
variationally invariant under the above mentioned diffeomorphisms of $Q$.
This invariance however leads to rather restrictive Noether identities
(\ref{s60}) and (\ref{s110}) which these Lagrangians and Hamiltonians
must satisfy, unless other fields are introduced.
In a different way, one can choose some subbundle of the fiber bundle
$Z_Q\to Q$ in order to avoid the above mentioned ambiguity between jets
of subbundles of $Z$ and jets of sections of $Z_Q\to Q$. Since such a
subbundle itself need not be a jet manifold of some fiber bundle, it is
a nonholonomic constraint. For
instance, this is the case of relativistic mechanics, phrased in terms of
four-velocities.
\section{Jets of submanifolds}
Given an $m$-dimensional smooth real manifold $Z$, a
$k$-order jet of $n$-dimensional submanifolds of $Z$ at a point
$z\in Z$ is defined as the equivalence class $j^k_zS$
of
$n$-dimensional imbedded submanifolds of $Z$ through $z$ which
are tangent to each other at $z$ with order $k\geq 0$.
Namely, two submanifolds
$i_S: S\hookrightarrow Z$, $i_{S'}:
S'\hookrightarrow Z$
through a point $z\in Z$ belong to the same equivalence class $j^k_zS$
iff the images of the $k$-tangent morphisms
\be
T^ki_S: T^kS\hookrightarrow T^kZ, \qquad T^ki_{S'}:
T^kS'\hookrightarrow T^kZ
\ee
coincide with each other.
The set $J^k_nZ=\op\bigcup_{z\in Z} j^k_zS$ of $k$-order jets is a
finite-dimensional real smooth manifold.
One puts $J^0_nZ =Z$. If $k>0$,
let $Y\to X$ be an $m$-dimensional
fiber bundle over an
$n$-dimensional base $X$ and $J^kY$ the $k$-order jet manifold of
sections of $Y\to X$ (or, shortly, the jet manifold of $Y\to X$). Given
an imbedding
$\Phi:Y\to Z$, there is the natural injection
\be
J^k\Phi: J^kY\to J^k_nZ, \qquad j^k_xs \mapsto [\Phi\circ
s]^k_{\Phi(s(x))},
\ee
where $s$ are sections of $Y\to X$. This injection defines a
chart on
$J^k_nZ$. These charts provide a manifold atlas of $J^k_nZ$.
Here, we restrict our consideration to first order jets of submanifolds.
There is obvious one-to-one correspondence
\mar{s10}\beq
\zeta: j^1_zS \mapsto V_{j^1S}\subset T_zZ \label{s10}
\eeq
between the jets $j^1_zS$ at a point $z\in Z$ and the $n$-dimensional
vector subspaces of the tangent space $T_zZ$ of $Z$ at $z$. It follows
that
$J^1_nZ$ is a fiber bundle
\mar{s3}\beq
\rho:J^1_nZ\to Z \label{s3}
\eeq
in Grassmann manifolds. It possesses the following coordinate atlas.
Let $\{(U;z^\m)\}$ be a coordinate atlas of $Z$.
Though $J^0_nZ=Z$, let us provide
$J^0_mZ$ with the atlas obtained by replacing every chart $(U,z^A)$ of
$Z$ with the
\be
{m\choose n}=\frac{m!}{n!(m-n)!}
\ee
charts on $U$ which
correspond to different partitions of $(z^A)$ in
collections of $n$ and $m-n$ coordinates
\mar{5.8}\beq
(U; x^a,y^i), \qquad a=1,\ldots,n, \qquad i=1,\ldots,m-n.\label{5.8}
\eeq
The transition functions between the coordinate charts (\ref{5.8}) of
$J^0_nZ$ associated with a coordinate chart
$(U,z^A)$ of $Z$ reduce to an exchange between
coordinates $x^a$ and $y^i$.
Transition functions between arbitrary coordinate charts of the
manifold $J^0_nZ$ take the form
\mar{5.26} \beq
x'^a = x'^a (x^b, y^k), \qquad y'^i = y'^i (x^b, y^k).
\label{5.26}
\eeq
Given an atlas of coordinate charts (\ref{5.8}) -- (\ref{5.26}) of the
manifold
$J^0_nZ$, the first order jet manifold $J^1_nZ$ is endowed with the
coordinate charts
\mar{5.31}\beq
(\rho^{-1}(U)=U\times\Bbb R^{(m-n)n}; x^a,y^i,y^i_a), \label{5.31}
\eeq
possessing the following transition functions.
With respect to the coordinates (\ref{5.31}) on the jet manifold $J^1_nZ$
and the induced fiber coordinates $(\dot x^a, \dot y^i)$ on the
tangent bundle $TZ$, the above mentioned correspondence $\zeta$
(\ref{s10}) reads
\be
\zeta: (y^i_a) \mapsto \dot x^a(\dr_a +y^i_a(j^1_zS)\dr_i).
\ee
It implies the relations
\mar{s0,1}\ben
&& y'^j_a= (\frac{\dr y'^j}{\dr y^k} y^k_b + \frac{\dr y'^j}{\dr x^b})
(\frac{\dr x^b}{\dr y'^i}y'^i_a + \frac{\dr x^b}{\dr x'^a}), \label{s0}\\
&& (\frac{\dr x^b}{\dr y'^i}y'^i_a + \frac{\dr x^b}{\dr x'^a})
(\frac{\dr x'^c}{\dr
y^k} y^k_b + \frac{\dr x'^c}{\dr x^b})=\dl^c_a,\label{s1}
\een
which jet coordinates $y^i_a$ must satisfy under coordinate
transformations (\ref{5.26}). Let consider a nondegenerate $n\times n$
matrix $M$ with the entries
\be
M^c_b=(\frac{\dr x'^c}{\dr
y^k}y^k_b + \frac{\dr x'^c}{\dr x^b}).
\ee
Then the relations (\ref{s1}) lead to the equalities
\be
(\frac{\dr x^b}{\dr y'^i} y'^i_a + \frac{\dr x^b}{\dr x'^a})=
(M^{-1})^b_a.
\ee
Hence, we obtain the
transformation law of first order jet coordinates
\mar{s2}\beq
y'^j_a=
( \frac{\dr y'^j}{\dr y^k} y^k_b+ \frac{\dr y'^j}{\dr x^b})
(M^{-1})^b_a.
\label{s2}
\eeq
For instance, these are the Lorentz transformation of three-velocities
in relativistic mechanics. In particular, if coordinate
transition functions $x'^a$ (\ref{5.26}) are independent of
coordinates $y^k$, the transformation law (\ref{s2}) comes to the familiar
transformations of jets of sections.
A glance at the transformations (\ref{s2}) shows that, in contrast
with a fiber bundle of jets of sections, the fiber bundle
(\ref{s3}) is not affine. In particular, one generalizes the notion of
a connection on fiber bundles and treat global sections of the jet
bundle (\ref{s3}) as preconnections
\cite{modu}. However, a global section of this bundle need not
exist (\cite{ste}, Theorem 27.18).
Given a coordinate chart (\ref{5.31}) of $J^1_nZ$, one can regard
$\rho^{-1}(U)\subset J^1_nZ$ as the first order jet manifold $J^1U$ of
sections of the fiber bundle
\mar{s11}\beq
U\ni (x^a,y^i)\to (x^a)\in U_X. \label{s11}
\eeq
The graded differential algebra $\cO^*(\rho^{-1}(U))$ of exterior forms
on
$\rho^{-1}(U)$ is generated by horizontal forms $dx^a$ and contact forms
$dy^i-y^i_adx^a$. Coordinate transformations (\ref{5.26}) and
(\ref{s2}) preserve the ideal of contact forms, but horizontal forms
are not transformed into horizontal forms, unless coordinate
transition functions $x'^a$ (\ref{5.26}) are independent of
coordinates $y^k$. Therefore, one can
develop first order Lagrangian formalism with a Lagrangian $L=\cL d^nx$
on a coordinate chart
$\rho^{-1}(U)$, but this Lagrangian fails to be globally defined on
$J^1_nZ$.
In order to overcome this difficulty, let us consider an above mentioned
product
$Z_Q=Q\times Z$ of $Z$ and an
$n$-dimensional real smooth manifold $Q$. We have a trivial fiber bundle
\mar{s12}\beq
\pi: Z_Q=Q\times Z\to Q, \label{s12}
\eeq
whose trivialization throughout holds fixed. This fiber bundle is
provided with an atlas of coordinate charts
\mar{s20}\beq
(U_Q\times U; q^\m,x^a,y^i), \label{s20}
\eeq
where $(U; x^a,y^i)$ are the above mentioned coordinate charts
$(\ref{5.8})$ of the manifold
$J^0_n Z$. The coordinate charts (\ref{s20}) possess
transition functions
\mar{s21}\beq
q'^\m=q^\m(q^\nu), \qquad x'^a = x'^a (x^b, y^k), \qquad y'^i = y'^i
(x^b, y^k). \label{s21}
\eeq
Let $J^1Z_Q$ be the first order jet manifold of the fiber bundle
(\ref{s12}). Since the trivialization (\ref{s12}) is fixed, it is a
vector bundle $\pi^1:J^1Z_Q\to Z_Q$ isomorphic to the tensor
product
\mar{s30}\beq
J^1Z_Q= T^*Q\op\ot_{Q\times Z} TZ \label{s30}
\eeq
of the cotangent bundle $T^*Q$ of $Q$ and the
tangent bundle $TZ$ of $Z$ over $Z_Q$.
Given a coordinate atlas
(\ref{s20}) - (\ref{s21}) of
$Z_Q$, the jet manifold $J^1Z_Q$ is endowed with the
coordinate charts
\mar{s14}\beq
((\pi^1)^{-1}(U_Q\times U)=U_Q\times U\times\Bbb R^{mn};
q^\m,x^a,y^i,x^a_\m, y^i_\m), \label{s14}
\eeq
possessing transition functions
\mar{s16}\beq
x'^a_\m=(\frac{\dr x'^a}{\dr
y^k}y^k_\nu + \frac{\dr x'^a}{\dr x^b}x^b_\nu )\frac{\dr q^\nu}{\dr
q'^\m}, \qquad y'^i_\m=(\frac{\dr
y'^i}{\dr y^k}y^k_\nu + \frac{\dr y'^i}{\dr x^b}x^b_\nu)\frac{\dr
q^\nu}{\dr q'^\m}. \label{s16}
\eeq
Relative to coordinates (\ref{s14}), the isomorphism (\ref{s30}) takes
the form
\be
(x^a_\m, y^i_\m) \to dq^\m\ot(x^a_\m \dr_a + y^i_\m \dr_i).
\ee
Obviously, a jet $(q^\m,x^a,y^i,x^a_\m, y^i_\m)$ of
sections of the fiber bundle (\ref{s12}) defines some jet of
$n$-dimensional subbundles of the manifold
$\{q\}\times Z$ through a point $(x^a,y^i)\in Z$ if an $m\times n$
matrix with the entries $x^a_\m, y^i_\m$ is of maximal rank $n$.
This property is
preserved under the coordinate transformations (\ref{s16}).
An element of $J^1Z_Q$ is called regular if it possesses this property.
Regular elements constitute an open subbundle of the jet bundle
$J^1Z_Q\to Z_Q$.
Since regular elements of $J^1Z_Q$ characterize jets of submanifolds of
$Z$, one hopes to describe the dynamics of
submanifolds of a manifold $Z$ as that of sections of the fiber bundle
(\ref{s12}). For this purpose, let us refine the relation
between elements of the jet manifolds $J^1_nZ$ and $J^1Z_Q$.
Let us consider the manifold product $Q\times J^1_nZ$. Of course, it is
a bundle over $Z_Q$. Given a coordinate atlas
(\ref{s20}) - (\ref{s21}) of
$Z_Q$, this product is endowed
with the coordinate charts
\mar{s13}\beq
(U_Q\times \rho^{-1}(U)=U_Q\times U\times\Bbb R^{(m-n)n};
q^\m,x^a,y^i, y^i_a), \label{s13}
\eeq
possessing transition functions (\ref{s2}).
Let us assign to an element $(q^\m,x^a,y^i, y^i_a)$ of the chart
(\ref{s13}) the elements $(q^\m,x^a,y^i,x^a_\m, y^i_\m)$ of the
chart (\ref{s14}) whose coordinates obey the relations
\mar{s17}\beq
\bx{y^i_a x^a_\m = y^i_\m.} \label{s17}
\eeq
These elements make up an $n^2$-dimensional vector space. The relations
(\ref{s17}) are maintained under coordinate transformations (\ref{s21})
and the induced transformations of the charts (\ref{s14}) and (\ref{s13})
as follows:
\be
&& y'^i_a x'^a_\m =
( \frac{\dr y'^i}{\dr y^k} y^k_c+
\frac{\dr y'^i}{\dr x^c}) (M^{-1})^c_a
(\frac{\dr x'^a}{\dr
y^k}y^k_\nu + \frac{\dr x'^a}{\dr x^b}x^b_\nu)\frac{\dr q^\nu}{\dr q'^\m}
= \\
&& \qquad ( \frac{\dr y'^i}{\dr y^k} y^k_c+
\frac{\dr y'^i}{\dr x^c}) (M^{-1})^c_a
(\frac{\dr x'^a}{\dr
y^k}y^k_b + \frac{\dr x'^a}{\dr x^b} )x^b_\nu\frac{\dr q^\nu}{\dr q'^\m}
=\\
&& \qquad (\frac{\dr y'^i}{\dr
y^k}y^k_b + \frac{\dr y'^i}{\dr x^b})x^b_\nu\frac{\dr q^\nu}{\dr q'^\m}=
(\frac{\dr y'^i}{\dr
y^k}y^k_\nu + \frac{\dr y'^i}{\dr x^b}x^b_\nu)\frac{\dr q^\nu}{\dr
q'^\m}= y'^i_\m.
\ee
Thus, one can associate
\mar{s25}\beq
\zeta': (q^\m,x^a,y^i, y^i_a) \mapsto \{(q^\m,x^a,y^i,x^a_\m, y^i_\m) \,
| \, y^i_a x^a_\m = y^i_\m\} \label{s25}
\eeq
to each element of the manifold $Q\times J^1_nZ$ an
$n^2$-dimensional vector space in the jet manifold $J^1Z_Q$. This
is a subspace of elements $x^a_\m dq^\m\ot(\dr_a + y^i_a\dr_i)$ of a
fiber of the tensor bundle (\ref{s30}) at a point
$(q^\m,x^a,y^i)$. This subspace always contains regular elements, e.g.,
whose coordinates $x^a_\m$ form a nondegenerate $n\times n$ matrix.
Conversely, given a regular element $j^1_zs$ of $J^1Z_Q$, there is a
coordinate chart (\ref{s14}) such that coordinates $x^a_\m$ of
$j^1_zs$ constitute a nondegenerate matrix, and $j^1_zs$ defines a unique
element of
$Q\times J^1_nZ$ by the relations
\mar{s31}\beq
\bx{y^i_a=y^i_\m(x^{-1})^\m_a.} \label{s31}
\eeq
For instance, this is the well-known relation between three- and
four-velocities in relativistic mechanics.
Thus, we have shown the following. Let $(q^\m,z^A)$ further be arbitrary
coordinates on the product
$Z_Q$ (\ref{s12}) and $(q^\m,z^A,z^A_\m)$ the corresponding coordinates
on the jet manifold $J^1Z_Q$. In these coordinates, an element of
$J^1Z_Q$ is regular if an $m\times n$ matrix with the entries $z^A_\m$ is
of maximal rank $n$.
\begin{prop} \label{s50} \mar{s50}
(i) Any jet of submanifolds through a point $z\in Z$ defines some (but
not unique) jet of sections of the fiber bundle $Z_Q$ (\ref{s12}) through
a point $q\times z$ for any $q\in Q$ in accordance with the relations
(\ref{s17}).
(ii) Any regular element of
$J^1Z_Q$ defines a unique element of the jet manifold $J^1_nZ$ by means
of the relations (\ref{s31}). However, nonregular elements of $J^1Z_Q$
can correspond to different jets of submanifolds.
(iii) Two elements $(q^\m,z^A,z^A_\m)$ and $(q^\m,z^A,z'^A_\m)$ of
$J^1Z_Q$ correspond to the same jet of submanifolds if $z'^A_\m= M^\nu_\mu
z^A_\nu$, where $M$ is some matrix, e.g., it
comes from a diffeomorphism of $Q$.
\end{prop}
Basing on this result, we can describe the
dynamics of $n$-dimensional submanifolds of a manifold $Z$ as that of
sections of the fiber bundle $Q\times Z\to Q$ for some $n$-dimensional
manifold $Q$.
\section{Lagrangian dynamics of submanifolds}
Let $Z_Q$ be a fiber bundle (\ref{s12}) coordinated by $(q^\m, z^A)$ with
transition functions $q'^\m(q^\nu)$ and $z'^A(z^B)$. Then the first
order jet manifold $J^1Z_Q$ of this fiber bundle is provided with
coordinates $(q^\m, z^A, z^A_\m)$ possessing transition functions
\be
z'^A_\m=\frac{\dr z'^A}{\dr z^B} \frac{\dr q^\nu}{\dr q'^\m} z^B_\nu.
\ee
Let us recall the notation of contact forms $\th^A=dz^A-z^A_\m dq^\m$,
operators of total derivatives
\be
d_\m=\dr_\m + z^A_\m\dr_A + z^A_{\m\nu}\dr_A^\nu,
\ee
the total differential $d_H (\f)= dq^\m\w d_\m(\f)$ acting on exterior
forms $\f$ on $J^1Z_Q$, and the horizontal projection $h_0(\th^A)= 0$.
A first order Lagrangian in Lagrangian formalism on a fiber bundle
$Z_Q\to Q$ is defined as a horizontal density
\mar{s40}\beq
L=\cL(z^A, z^A_\m) \om, \qquad \om=dq^1\w\cdots \w dq^n, \label{s40}
\eeq
on the jet manifold $J^1Z_Q$. The corresponding Euler--Lagrange operator
reads
\mar{s41}\beq
\dl L= \cE_A dz^A\w \om, \qquad \cE_A= \dr_A\cL - d_\m \dr_A^\m\cL.
\label{s41}
\eeq
It yields the Euler--Lagrange equations
\mar{s90}\beq
\cE_A= \dr_A\cL - d_\m \dr_A^\m\cL =0. \label{s90}
\eeq
Let $u=u^\m\dr_\m + u^A\dr_A$ be a vector field on $Z_Q$. Its jet
prolongation onto $J^1Z_Q$ reads
\mar{s52}\beq
u= u^\m\dr_\m + u^A\dr_A + (d_\m u^A -z^A_\nu d_\m u^\nu)\dr_A^\m.
\label{s52}
\eeq
It admits the vertical splitting
\mar{s53}\beq
u= u_H + u_V= u^\m d_\m +
[(u^A-u^\nu z^A_\nu)\dr_A + d_\m(u^A -z^A_\nu u^\nu)\dr_A^\m].
\label{s53}
\eeq
The Lie derivative $\bL_uL$ of a Lagrangian $L$ along a vector field
$u$ obeys the first variational formula
\mar{s42}\beq
\bL_uL = u_V\rfloor \dl L + d_H(h_0(u\rfloor H_L))=((u^A-u^\m
z^A_\m) \cE_A + d_\m
\gJ^\m)\om,
\label{s42}
\eeq
where
\mar{s43}\beq
H_L=L +\dr_A^\m\cL\th^A\w \om_\m,
\qquad \om_\m=\dr_\m\rfloor \om, \label{s43}
\eeq
is the Poincar\'e--Cartan form and
\mar{s44}\beq
\gJ=\gJ^\m\om_\m=( \dr^\m_A\cL(u^A- u^\nu z^A_\nu) + u^\m\cL)\om_\m
\label{s44}
\eeq
is the Noether current. A vector field $u$ is called a variational
symmetry of a Lagrangian $L$ if the Lie derivative (\ref{s42}) is
$d_H$-exact, i.e., $\bL_u L=d_H\si$. In this case, there is the weak
conservation law $0\ap d_H(\gJ-\si)$
on the shell (\ref{s90}).
One can show that a vector field
$u$ (\ref{s52}) is a variational symmetry only if it is projected onto
$Q$, i.e., its components $u^\m$ are functions on $Q$, and iff its
vertical part $u_V$ (\ref{s53}) is a variational symmetry.
In a general setting, one deals with generalized vector fields
$u$ depending on parameter functions $\xi^r(q^\nu)$, their derivatives
$\dr_{\la_1\ldots\la_k}\xi^r$, and higher order jets
$z^A_{\la_1\ldots\la_k}$ \cite{cmp04,jpa05}. A vertical variational
symmetry depending on parameters is called a gauge symmetry. Here we
restrict our consideration to gauge symmetries
$u$ which are linear in parameters and their first derivatives, i.e.,
\mar{s55}\beq
u= u^A\dr_A + d_\m u_A\dr^\m_A, \qquad u^A= u^A_r\xi^r +
u^{A,\m}_r\dr_\m\xi^r, \label{s55}
\eeq
where $u^A$ are functions of $q^\m$, $z^B$ and the jets
$z^B_{\m_1\ldots\m_k}$ of bounded jet order $k<N$. By virtue of the
Noether second theorem
\cite{jmp05,jpa05,jmp05a},
$u$ (\ref{s55}) is a gauge
symmetry of a Lagrangian $L$ (\ref{s40}) iff the variational derivatives
$\cE_A$ (\ref{s41}) of
$L$ obey the Noether identities
\mar{s56}\beq
(u^A_r-d_\m u^{A,\m}_r)\cE_A - u^{A,\m}_r d_\m\cE_A=0. \label{s56}
\eeq
For instance, let us consider an arbitrary vector field
$u=u^\m(q^\nu)\dr_\m$ on
$Q$. It is an infinitesimal generator of a one-parameter group of local
diffeomorphisms of $Q$. Since $Z_Q\to Q$ is a trivial bundle, this
vector field gives rise to a vector field $u=u^\m\dr_\m$ on $Z_Q$, and its
jet prolongation (\ref{s52}) onto
$J^1Z_Q$ reads
\mar{s59}\beq
u= u^\m \dr_\m - z^A_\nu\dr_\m u^\nu \dr_A^\m= u^\m d_\m +[-
u^\nu z^A_\nu\dr_A - d_\m (u^\nu z^A_\nu)\dr_A^\m].
\label{s59}
\eeq
One can regard it as a generalized vector field depending on parameter
functions $u^\m(q^\nu)$. In accordance with Proposition 1,
it seems reasonable to require that, in order to describe jets of
submanifolds of $Z$, a Lagrangian $L$ on $J^1Z_Q$ is independent on
coordinates of $Q$ and must be variationally invariant under $u$
(\ref{s59}) or, equivalently, its vertical part
\be
u_V= -
u^\nu z^A_\nu\dr_A - d_\m(u^\nu z^A_\nu)\dr_A^\m.
\ee
Then the variational derivatives of this Lagrangian obey
irreducible Noether identities (\ref{s56}) which read
\mar{s60}\beq
\bx{z^A_\nu\cE_A=0.} \label{s60}
\eeq
These Noether identities are rather restrictive, unless other fields are
introduced.
In order to extend Lagrangian formalism on $Z_Q$ to other
fields, one can use the following two constructions:
(i) a bundle product
$Z_Q\times Z'$ of $Z_Q$ and some bundle $Z'\to Q$ bundle over
$Q$ (e.g., a tensor bundle $\op\ot^kTQ\op\ot^rT^*Q$), (ii) a bundle
$E\to Z$ and its pull-back $E_Q$ onto $Q\times Z$ which is a composite
bundle
\mar{s61}\beq
E_Q=Q\times E\to Z_Q \to Q. \label{s61}
\eeq
Let $E\to Z$ be provided with bundle coordinates $(z^A, s^i)$. Its
pull-back $E_Q$ onto $Z_Q$ possesses coordinates $(q^\m,z^A,s^i)$.
Accordingly, the pull-back
$Q\times J^1E$ of the first order jet manifold
$J^1E$ of
$E\to Z$ onto
$Z_Q$ is endowed with coordinates $(q^\m,z^A, s^i, s^i_A)$. It is a
subbundle $Q\times J^1E\subset J^1E_Q$ of the first order jet manifold
$J^1E_Q\to Z_Q$ of the fiber bundle $E_Q\to Z_Q$. This subbundle consists
of jets of sections of $E_Q\to Z_Q$ which are the pull-back of sections of
$E\to Z$. Given a composite fiber bundle (\ref{s61}), there is the
canonical bundle morphism
\mar{s62}\beq
\g: J^1Z_Q\op\times_{Z_Q} J^1E_Q \op\to_{Z_Q} J^1_QE_Q \label{s62}
\eeq
of the bundle product of jet manifolds $J^1Z_Q$, $J^1E_Q$ of the bundles
$Z_Q\to Q$, $E_Q\to Z_Q$ to the first order jet manifold $J^1_QE_Q$ of
the fiber bundle $E_Q\to Q$ \cite{book,sau,book00}. The jet
manifold $J^1_QE_Q$ is coordinated by $(q^\m,z^A,s^i,z^A_\m,s^i_\m)$.
Restricted to
$Q\times J^1E\subset J^1E_Q$, the morphism (\ref{s62}) takes the
coordinate form
\mar{s63}\beq
(z^A_\m,s_\m^i)=\g(z^A_\m,s^i_A)=(z^A_\m,s_A^iz^A_\m). \label{s63}
\eeq
Due to the morphism (\ref{s63}), any connection
\mar{s64}\beq
\G=dz^A\ot(\dr_A + \G^i_A(z^B,s^j)\dr_i) \label{s64}
\eeq
on a fiber bundle $E\to Z$ yields the covariant derivative
\mar{s65}\beq
D_\m s^i=s^i_\m - \G^i_A(z^B,s^j)z^A_\m \label{s65}
\eeq
on the composite bundle $E_Q\to Q$.
Given a fiber bundle $Z_Q\times Z'$ or a composite fiber bundle
$E_Q$ (\ref{s61}), an extended Lagrangian is defined on the
jet manifolds $J^1Z_Q\times J^1Z'$ or $J^1_QE_Q$, respectively. For
instance, any horizontal $n$-form
\be
\frac1{n!}\f_{A_1\ldots A_n}dz^{A_1}\w\cdots\w dz^{A_n}
\ee
on $E\to Z$ yields a horizontal density
\be
\frac1{n!}\f_{A_1\ldots A_n}z^{A_1}_{\m_1}\cdots
z^{A_n}_{\m_n}dq^{\m_1}\w\cdots\w dq^{\m_n}
\ee
on $J^1_QE_Q\to Q$ which may contribute to a Lagrangian (see,
e.g., a relativistic particle in
the presence of an electromagnetic field).
\section{Hamiltonian dynamics of submanifolds}
Here, we follow
polysymplectic Hamiltonian formalism which aims to describe
field systems with nonregular Lagrangians
\cite{book,jpa99,sard94,book95}. Lagrangian and polysymplectic
Hamiltonian formalisms are equivalent in the case of hyperregular
Lagrangians, but a nonregular Lagrangian admits different associated
Hamiltonians, if any. At the same time, there is a comprehensive relation
between these formalisms in the case of almost-regular Lagrangians.
Given a fiber bundle $Z_Q$ (\ref{s12}) and its vertical cotangent bundle
$V^*Z_Q$, let us consider the fiber bundle
\mar{s70}\beq
\Pi=V^*Z_Q\w (\op\w^{n-1} T^*Q). \label{s70}
\eeq
It plays the role of a momentum phase space
of covariant Hamiltonian field theory. Given coordinates
$(q^\m,z^A)$ on
$Z_Q$, this fiber bundle is coordinated by $(q^\m, z^A, p^\m_A)$, where
$p^\m_A$ are treated as coordinates of momenta. It is provided with the
canonical polysymplectic form
\mar{s73}\beq
\Om_\Pi= dp_A^\m\w dz^A\ot \dr_\m. \label{s73}
\eeq
Every Lagrangian $L$ on the jet manifold
$J^1Z_Q$ yields the Legendre map
\mar{s71}\beq
\wh L: J^1Z_Q\op\to_{Z_Q} \Pi, \qquad p^\m_A\circ\wh L =\dr^\m_A\cL,
\label{s71}
\eeq
whose range $N_L=\wh L(J^1Z_Q)$ is called the Lagrangian constraint
space. A Lagrangian $L$ is called hyperregular (resp. regular), if the
Legendre map (\ref{s71}) is a diffeomorphism (resp. local diffeomorphism,
i.e., of maximal rank). A Lagrangian
$L$ is said to be almost-regular if the Lagrangian constraint space is a
closed imbedded subbundle
$i_N:N_L\to\Pi$ of the Legendre bundle $\Pi\to Z_Q$ and the surjection
$\wh L:J^1Z_Q\to N_L$ is a fibered manifold possessing connected fibers.
A multisymplectic momentum phase space is the homogeneous Legendre
bundle
\be
\ol \Pi=T^*Z_Q\w (\op\w^{n-1} T^*Q),
\ee
coordinated by $(q^\m, z^A, p^\m_A, p)$. It is endowed with the canonical
multisymplectic form
\be
\Xi=p\om + p^\m_A dz^A\w\om_\m.
\ee
There is a trivial one-dimensional bundle $\ol\Pi\to \Pi$.
Then a Hamiltonian $\cH$ on the momentum phase space $\Pi$ (\ref{s70}) is
defined as a section
$p=-\cH$ of this fiber bundle. The pull-back of the multisymplectic form
$\Xi$ onto
$\Pi$ by a Hamiltonian $\cH$ is a Hamiltonian form
\mar{s75}\beq
H= p^\m_Adz^A\w\om_\m -\cH\om \label{s75}
\eeq
on $\Pi$. The corresponding Hamilton equations with respect to the
polysymplectic form $\Om_\Pi$ (\ref{s73}) read
\be
z^A_\m-\dr^A_\m\cH=0, \qquad -p^\m_{\m A}- \dr_A\cH=0.
\ee
A key point is that these Hamilton equations coincide with
the Euler--Lagrange equations of the first order Lagrangian
\mar{s77}\beq
L_H= (p^\m_Az^A_\m - \cH) \label{s77}
\eeq
on the jet manifold $J^1\Pi$ of $\Pi\to Q$. Indeed, its variational
derivatives are
\mar{s101}\beq
\cE^A_\m=z^A_\m-\dr^A_\m\cH, \qquad \cE_A=-p^\m_{\m A}- \dr_A\cH.
\label{s101}
\eeq
Any Hamiltonian form $H$ (\ref{s75}) on $\Pi$ yields the Hamiltonian map
\mar{s80}\beq
\wh H: \Pi\op\to_{Z_Q} J^1Z_Q, \qquad z^A_\m\circ \wh H= \dr^A_\m\cH.
\label{s80}
\eeq
A Hamiltonian $\cH$ on $\Pi$ is said to be associated to a Lagrangian $L$
on $J^1Z_Q$ if it satisfies the relations
\mar{s81,2}\ben
&& p^\m_A=\dr^\m_A\cL(q^\nu, z^B, z^B_\la=\dr^B_\la\cH), \label{s81}\\
&& p^\m_A\dr_\m^A\cH -\cH=\cL(q^\nu, z^B, z^B_\la=\dr^B_\la\cH).
\label{s82}
\een
If an associated Hamiltonian $\cH$ exists, the Lagrangian constraint space
$N_L$ is given by the coordinate equalities (\ref{s81}).
The relation between Lagrangian and polysymplectic Hamiltonian
formalisms is based on the following facts.
(i) Let a Lagrangian $L$ be
almost regular, and let us assume that it admits an associated
Hamiltonian $\cH$, which however need not be unique, unless $L$ is
hyperregular. In
this case, the Poincar\'e--Cartan form
$H_L$ (\ref{s43}) is the pull-back $H_L=\wh L^*H$ of the Hamiltonian form
$H$ (\ref{s75}) for any associated Hamiltonian $\cH$. Note that a local
associated Hamiltonian always exists. The Poincar\'e--Cartan form $H$ is
a Lepagean equivalent both of the original Lagrangian $L$ on $J^1Z_Q$ and
the Lagrangian
\be
\ol L=(\cL + (\wh z^A_\m - z^A_\m)\dr^\m_A\cL)\om
\ee
on the repeated jet manifold $J^1J^1Z_Q$. Its Euler--Lagrange equations
are the Cartan equations for $L$. Any solution of the Euler--Lagrange
equations (\ref{s90}) for $L$ is also a solution of the Cartan equations.
Furthermore, Euler--Lagrange equations and the Cartan equations are
equivalent in the case of a regular Lagrangian.
(ii) If a Lagrangian $L$ is almost regular, all
associated Hamiltonian forms $H$ coincide with each other on the
Lagrangian constraint space $N_L$, and define the constrained Lagrangian
$L_N= h_0(i_N^*H)$ on the jet manifold $J^1N_L$ of the fiber bundle
$N_L\to Q$. The Euler--Lagrange equations for this Lagrangian are called
the constrained Hamilton equations. In fact, the Lagrangian $L_H$
(\ref{s77}) is defined on the bundle product
\mar{s100}\beq
\Pi\op\times_{Z_Q} J^1Z_Q, \label{s100}
\eeq
and the constrained Lagrangian $L_N$ is the restriction of
$L_H$ to $N_L\times J^1Z_Q$.
As a result, one can show that a section $\ol
S$ of the jet bundle $J^1Z_Q\to Q$ is a solution of the Cartan equations
for $L$ iff $\wh L\circ \ol S$ is a solution of the constrained Hamilton
equations. In particular, any solution $r$ of the constrained Hamilton
equations provides the solution $\ol S=\wh H\circ r$ of the Cartan
equations.
Turn now to symmetries of a Lagrangian $L_H$ (\ref{s77}). Any vector
field $u$ on $Z_Q$ gives rise to the vector field
\be
u_\Pi= u^\m\dr_\m + u^A\dr_A +(-\dr_A u^B p^\m_B -\dr_\la u^\la p^\m_A
+\dr_\la u^\m p^\la_A)\dr^A_\m
\ee
onto the Legendre bundle $\Pi$. Then we obtain its prolongation
\be
u_\Pi= u^\m\dr_\m + u^A\dr_A + (d_\m u^A -z^A_\nu d_\m
u^\nu)\dr_A^\m + (-\dr_A u^B p^\m_B -\dr_\la u^\la p^\m_A +\dr_\la u^\m
p^\la_A)\dr^A_\m
\ee
onto the product (\ref{s100}). It is a variational symmetry of the
Lagrangian $L_H$ if the Lie derivative $\bL_{u_\Pi}L_H$ is $d_H$-exact.
For instance, let $u=u^\m\dr_\m$ be an arbitrary vector field on $Q$.
Since $Z_Q\to Q$ is a trivial bundle, this
vector field gives rise to a vector field $u=u^\m\dr_\m$ on $Z_Q$ whose
lift onto the Legendre bundle $\Pi$ is
\be
u_\Pi= u^\m\dr_\m + (-\dr_\la u^\la p^\m_A +\dr_\la u^\m
p^\la_A)\dr^A_\m.
\ee
Then we obtain its prolongation
\be
u_\Pi=u^\m\dr_\m - z^A_\nu\dr_\m u^\nu\dr^\m_A + (-\dr_\la u^\la p^\m_A
+\dr_\la u^\m p^\la_A)\dr^A_\m
\ee
onto the product (\ref{s100}), and take its vertical part
\mar{s105}\beq
u_V= -u^\nu z^A_\nu\dr_A - d_\m(u^\nu z^A_\nu)\dr_A^\m + (-\dr_\la u^\la
p^\m_A +\dr_\la u^\m p^\la_A -u^\nu p_{\nu A}^\m)\dr^A_\m. \label{s105}
\eeq
Let us regard it as a generalized vector field dependent on parameter
functions $u^\m(q)$. In accordance with Proposition 1, let us require
that a Lagrangian $L_H$ is independent on coordinates on $Q$ and
possesses the gauge symmetry $u_V$ (\ref{s105}). Then its
variational derivatives (\ref{s101}) of $L_H$ obey
the Noether identities
\be
z^A_\nu \cE_A+ p^\m_{\m A} \cE^A_\nu + p^\m_A(d_\m\cE_\nu^A -
d_\nu\cE^A_\m) =0,
\ee
which reduce to rather restrictive conditions
\mar{s110}\beq
\bx{\dl^\m_\nu \cH =(n-1) p^\m_A\dr^A_\nu\cH}
\label{s110}
\eeq
which a Hamiltonian $\cH$ must satisfy. For instance, $\cH=0$ if $n=1$.
In this case, momenta are scalars relative to transformations of $q$
and, therefore, no function of them is a density with respect to these
transformations.
\section{Example. $n=1,2$}
Given an $m$-dimensional manifold $Z$ coordinated by $(z^A)$, let us
consider the jet manifold $J^1_1Z$ of its one-dimensional submanifolds.
Let us provide $Z=Z^0_1$ with coordinates $(x^0=z^0, y^i=z^i)$
(\ref{5.8}). Then the jet manifold $J^1_1Z$ is endowed with coordinates
$(z^0,z^i,z^i_0)$ possessing transition functions (\ref{5.26}),
(\ref{s2}) which read
\mar{s120}\beq
z'^0=z'^0(z^0,z^k), \quad z'^0=z'^0(z^0,z^k), \quad z'^i_0=
(\frac{\dr z'^i}{\dr z^j} z^j_0 + \frac{\dr z'^i}{\dr z^0} )
(\frac{\dr z'^0}{\dr z^j} z^j_0 + \frac{\dr z'^0}{\dr z^0} )^{-1}.
\label{s120}
\eeq
A glance at the transformation law (\ref{s120}) shows that $J^1_1Z\to Z$
is a fiber bundle in projective spaces.
For instance, put $Z=\Bbb R^4$ whose Cartesian coordinates are subject
to the Lorentz transformations
\mar{s122}\beq
z'^0= z^0{\rm ch}\al - z^1{\rm sh}\al, \qquad z'^'= -z^0{\rm sh}\al +
z^1{\rm ch}\al, \qquad z'^{2,3} = z^{2,3}. \label{s122}
\eeq
Then $z'^i$ (\ref{s120}) are exactly the Lorentz transformations
\be
z'^1_0=\frac{ z^1_0{\rm ch}\al -{\rm sh}\al}{ - z^1_0{\rm
sh}\al+ {\rm ch}\al} \qquad
z'^{2,3}_0=\frac{z^{2,3}_0}{ - z^1_0{\rm
sh}\al + {\rm ch}\al}
\ee
of three-velocities in relativistic mechanics \cite{sard98,book98}.
Let us consider a one-dimensional manifold $Q=\Bbb R$
and the product $Z_Q=\Bbb R\times Z$. Let $\Bbb R$ be provided with a
Cartesian coordinate $\tau$ possessing transition function $\tau'=\tau +
{\rm const}$, unless otherwise stated.
Then the jet manifold
$J^1Z_Q$ of the fiber bundle $\Bbb R\times Z\to Z$ is endowed with the
coordinates $(\tau, z^0, z^i, z^0_\tau, z^i_\tau)$ with the transition
functions
\mar{s123}\beq
z'^0_\tau=\frac{\dr z'^0}{\dr z^k} z^k_\tau + \frac{\dr z'^0}{\dr z^0}
z^0_\tau, \qquad
z'^i_\tau=\frac{\dr z'^i}{\dr z^k} z^k_\tau + \frac{\dr z'^i}{\dr z^0}
z^0_\tau. \label{s123}
\eeq
A glance at this transformation law shows that, unless
nonadditive transformations of
$\tau$ are considered, there is
an isomorphism
\mar{s121}\beq
J^1_1Z_Q =VZ_Q=\Bbb R\times TZ \label{s121}
\eeq
of the jet manifold $J^1_1Z_Q$ to the vertical tangent bundle $VZ_Q$ of
$Z_Q\to \Bbb R$ which, in turn, is a product of
$\Bbb R$ and the tangent bundle $TZ$ of $Z$.
Returning to the example of $Z=\Bbb R^4$ and Lorentz transformations
(\ref{s122}), one easily observed that transformations (\ref{s123}) are
transformations of four-velocities in relativistic mechanics where $\tau$
is a proper time.
Let us consider coordinate charts $(U';\tau, z^0,z^i,z^i_0)$ and
$(U'';\tau, z^0,z^i,z^0_\tau, z^i_\tau)$ of the manifolds $\Bbb R\times
J^1_1Z$ and
$J^1Z_Q$ over the same coordinate chart $(U;\tau, z^0,z^i)$ of $Z_Q$.
Then one can associate to each element $(\tau, z^0,z^i,z^i_0)$ of
$U'\subset \Bbb R\times
J^1_1Z$ the elements of $U''\subset J^1Z_Q$ which obey the relations
\mar{s125}\beq
z^i_0 z^0_\tau= z^i_\tau \label{s125}
\eeq
and, in particular, the relations
\mar{s126}\beq
z^i_0= \frac{z^i_\tau}{z^0_\tau}, \qquad z^0_\tau\neq 0. \label{s126}
\eeq
Given a point $(\tau,z)\in \Bbb R\times
Z$, the relations (\ref{s125}) -- (\ref{s126}) are exactly the
correspondence between elements of a one-dimensional vector subspace of
the tangent space $T_zZ$ and the corresponding element of the projective
space of these subspaces.
In the above mentioned example of relativistic
mechanics, the relations (\ref{s125}) -- (\ref{s126}) are familiar
equalities between three- and four-velocities. It should be emphasized
that, in relativistic mechanics, one avoids the ambiguity between
three- and four-velocities by means of the nonholonomic constraint
\mar{s131}\beq
(z^0_\tau)^2- \op\sum_i(z^i_\tau)^2 =1. \label{s131}
\eeq
In a general setting, Lagrangian formalism on the jet manifold $J^1Z_Q$
can be developed if a Lagrangian $L$ is independent of $\tau$, and it is
variationally invariant under transformations reparametrizations
$\tau'(\tau)$, i.e., its Euler--Lagrange operator obeys the Noether
identity
\mar{s130}\beq
z^A_\tau\cE_A=0. \label{s130}
\eeq
For instance, let $Z$ be a locally affine manifold, i.e., a toroidal
cylinder $\Bbb R^{m-k}\times T^k$. Its tangent bundle can be provided
with a constant nondegenerate fiber metric $\eta_{AB}$. Then
\mar{s132}\beq
L=(\eta_{AB}z^A_\tau z^B_\tau)^{1/2}d\tau \label{s132}
\eeq
is a
Lagrangian on $J^1Z_Q$. It is easily justified that this Lagrangian
satisfies the Noether identity (\ref{s130}). Furthermore, given a one-form
$\cA_Bdz^B$ on $Z$, one can consider the Lagrangian
\label{s133}\beq
L'=[(\eta_{AB}z^A_\tau z^B_\tau)^{1/2} - \cA_Bz^B_\tau]d\tau, \label{s133}
\eeq
which also obeys the Noether identity (\ref{s130}).
In relativistic mechanics, the Euler--Lagrange equations of the
Lagrangians $L$ (\ref{s132}) and $L'$ (\ref{s133}) restricted to the
constraint space (\ref{s131}) restart the familiar equations of motion of
a free relativistic particle particle and a relativistic particle in the
presence of an electromagnetic field $\cA$.
As was mentioned above, no Hamiltonian obeys the Noether identities
(\ref{s110}) if $n=1$. However, Hamiltonian relativistic
mechanics can be developed in the framework of Hamiltonian theory of
mechanical systems with nonholonomic constraints
\cite{book98,sard99,sard03}. A key is that the constraint condition
(\ref{s131}) is not preserved under transformations of $\tau$, and a
Hamiltonian of a mechanical system with this constraint need not
satisfy the Noether identities (\ref{s110}).
In comparison with the case of one-dimensional submanifolds, a
description of the Lagrangian and Hamiltonian dynamics of two-dimensional
submanifolds follows general theory of $n$-dimensional submanifolds. This
is the case of classical string theory
\cite{scherk,hatf,polch}.
For instance, let $Z$ be again an $m$-dimensional locally affine manifold,
i.e., a toroidal cylinder $\Bbb R^{m-k}\times T^k$,
and let $Q$ be a
two-dimensional manifold. As was mentioned above, the tangent bundle
of $Z$ can be provided with a
constant nondegenerate fiber metric
$\eta_{AB}$. Let us consider the $2\times 2$ matrix with the
entries
\be
h_{\m\nu}=\eta_{AB} z^A_\m z^B_\nu.
\ee
Then its determinant provides a
Lagrangian
\mar{s140}\beq
L=(\det h)^{1/2} d^2q =([\eta_{AB} z^A_1 z^B_1] [\eta_{AB} z^A_2
z^B_2]- [\eta_{AB} z^A_1 z^B_2]^2 )^{1/2} d^2q \label{s140}
\eeq
on the jet manifold $J^1Z_Q$ (\ref{s30}). This is the well known
Nambu--Goto Lagrangian of string theory. It satisfies the Noether
identities (\ref{s60}). Let
\be
F=\frac12 F_{AB} dz^A\w dz^B
\ee
be a two-form on a manifold $Z$.
Then
\be
F=\frac12 F_{AB} z^A_\m z^B_\nu dq^\m\w dq^\nu
\ee
is a horizontal density on $J^1Z_Q$ which can be treated as an
interaction term of submanifolds and an external classical field $F$ in
a Lagrangian.
Turn now to Hamiltonian theory of two-dimensional submanifolds on the
momentum phase space $\Pi$ (\ref{s70}). In this case, the Noether
identities (\ref{s110}) take the form
\mar{s145}\beq
\dl^\m_\nu\cH=p^\m_A\dr^A_\nu\cH. \label{s145}
\eeq
For instance, let $Z$ be the above mentioned toroidal cylinder whose
cotangent bundle is provided with a constant nondegenerate fiber metric
$\eta_{AB}$. Let us consider the $2\times 2$ matrix with the
entries
\be
H^{\m\nu}=\eta^{AB} p_A^\m p_B^\nu.
\ee
Then its determinant provides a Hamiltonian
\be
\cH=(\det H)^{1/2} d^2q =([\eta^{AB} p_A^1 p_B^1] [\eta^{AB} p_A^2
p_B^2]- [\eta^{AB} p_A^1 p_B^2]^2 )^{1/2} d^2q
\ee
on the momentum phase space $\Pi$ which satisfies the Noether identities
(\ref{s145}. This Hamiltonian is associated to the Lagrangian
(\ref{s140}). | 203,228 |
\begin{document}
\title[On Foreman's maximality principle]{On Foreman's maximality principle}
\author[Mohammad Golshani and Yair Hayut]{Mohammad
Golshani and Yair Hayut}
\thanks{The first author's research was in part supported by a grant from IPM (No. 91030417).}
\begin{abstract}
In this paper we consider Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We observe that it is consistent that every $c.c.c.$ forcing adds a real and that for every uncountable regular cardinal $\kappa$, every $\kappa$-closed forcing of size $2^{<\kappa}$ collapses some cardinal.
\end{abstract}
\maketitle
\section{introduction}
Foreman's maximality principle \cite{foreman-magidor-shelah} says that any non-trivial forcing notion either adds a new real or collapses some cardinal. The consistency of this principle was asked by Foreman-Magidor-Shelah \cite{foreman-magidor-shelah}, who showed that if $0^\sharp$ exists, then any non-trivial constructible forcing notion adds a new real over $V$ (see also \cite{stanley}, where a generalization of this result is proved).
In this paper we consider the following two consequences of Foreman's maximality principle, and prove some consistency results related to them:
\begin{enumerate}
\item Any non-trivial $c.c.c.$ forcing notion adds a new real.
\item For every uncountable regular cardinal $\kappa$, every $\kappa$-closed forcing of size $2^{<\kappa}$ collapses some cardinal.
\end{enumerate}
We show that statement $(1)$ is equivalent to the assertion ``there are no Souslin trees'', and hence by Solovay-Tennenbaum \cite{solovay-tennenbaum}, it is consistent that all non-trivial $c.c.c.$ forcing notion add a new real.
We also consider statement $(2)$, and prove that it is consistent, relative to the existence of a strong cardinal, that for all uncountable cardinals $\kappa,$ all $\k$-closed (and in fact all $\k$-strategically closed) forcing notions of size $\leq 2^{<\kappa}$ collapse $2^{<\k}.$ In such a model $GCH$ must fail everywhere, and hence we need some large cardinals to get the result. In the opposite direction, we build, assuming some large cardinals, a model in which $GCH$ fails everywhere, but for each infinite cardinal $\k,$ there exists a $\k^+$-closed forcing notion of size $2^\k$ which preserves all cardinals. Our work extends an earlier work of Foreman and Woodin \cite{foreman-woodin} by reducing their use of a supercompact cardinal to a strong cardinal.
In order to avoid trivialities, in this paper, the phrase ``forcing notion" is used only for separative non-atomic forcing notions.
\section{Consistency of any $c.c.c.$ forcing notion adding a new real}
In this section we consider statement $(1)$, and prove its consistency.
\begin{theorem}
Souslin hypothesis (SH) holds iff any non-trivial $c.c.c.$ forcing notion adds a new real.
\end{theorem}
\begin{proof}
One direction is trivial, since a Souslin tree, considered as a forcing notion, is $c.c.c.$ and adds no new reals.
For the other direction suppose there is a non-trivial $c.c.c.$ forcing notion $\mathbb{P}$ which adds no new reals. Let $\mathbb{B}=R.O(\mathbb{P})$ be the boolean completion of $\mathbb{P}$. $\mathbb{B}$ is a $c.c.c.$ complete Boolean algebra which is $(\omega, \omega)$-distributive, hence it is in fact $(\omega, \infty)$-distributive, thus it is a Souslin algebra, which implies the existence of a Souslin tree (see \cite{jech}). Hence SH fails.
\end{proof}
\begin{remark}
The above theorem is also proved by Shelah \cite{shelah2}, by completely different methods.
\end{remark}
As a corollary of the above theorem, and results of Solovay-Tennenbaum \cite{solovay-tennenbaum} (see also \cite{shelah1}), we have the following.
\begin{corollary}
It is consistent that any non-trivial $c.c.c.$ forcing notion adds a new real.
\end{corollary}
\section{Consistently, for every uncountable $\kappa,$ every $\kappa$-closed forcing of size $2^{<\kappa}$ collapses some cardinal}
In this section, we consider statement $(2),$ and prove the following consistency result.
\begin{theorem}
Assuming the existence of an $\aleph_{\kappa^{++}}$-strong cardinal $\kappa$, it is consistent that for all uncountable cardinals $\lambda,$ any non-trivial $\lambda$-closed forcing notion of size $\leq 2^{<\lambda}$ collapses some cardinal.
\end{theorem}
\begin{remark}
The conclusion of the theorem implies $GCH$ fails everywhere, so some very large cardinals are needed for the theorem.
\end{remark}
\begin{proof}
To prove the theorem, we need two lemmas, which are of some independent interest.
\begin{lemma}\label{lem: singular continuum collapsing} if $2^{\kappa}$ is singular, then every non-trivial $\kappa^+$-closed forcing of size $2^{\kappa}$ collapses $2^{\kappa}$.
\end{lemma}
\begin{proof}
Let $\mathbb{P}$ be a non-trivial $\kappa^+$-closed forcing notion of size $2^{\kappa}$. Then forcing with $\mathbb{P}$ adds a new sequence $\tau$ of ordinals of size $\lambda < 2^{\kappa}$ (since the minimal such $\lambda$ must be regular). We will encode $2^\kappa$ into $\tau$.
If we choose $\lambda$ to be minimal, every initial segment of $\tau$ is in $V$. Now, we can define a function $F\in V$ from all possible initial segments of $\tau$ onto $^{\kappa}2$, such that for every $p\in \mathbb{P}$ and every $x\in$~$^{\kappa}2$ there is $q\leq p$ and $\beta < \lambda$ such that $q\Vdash \tau \restriction \beta = \check{a}$ and $F(a)=x$.
Let $\langle (p_i, x_i) : i < 2^{\kappa}\rangle$ enumerate $\mathbb{P}\times$~$ ^{\kappa}2$. For every $\alpha < 2^{\kappa}$, we use the $\kappa^+$-closure of $\mathbb{P}$ and the fact that $\tau \notin V$ in order to find $q \leq p_\alpha$ such that $q \neq p_i$ for every $i < \alpha$, $q\Vdash \tau \restriction \beta = \check{a}$ (for some $\beta$) and $F(a)$ is not determined yet, and set $F(a)=x_\alpha$.
Let $p = p_\alpha \in \mathbb{P}$. We start by building a tree of $2^{<\kappa}$ incompatible conditions $q_s,\, s\in$~$^{<\kappa}2$ such that $q_\emptyset = p$ and for every $s\in$~$^{<\kappa}2$, $q_{s^\smallfrown (0)}, q_{s^\smallfrown(1)} \leq q_s$, there is $\beta_s < \lambda$ such that $q_{s^\smallfrown(i)} \Vdash \tau \restriction \check\beta_s = \check{a}_{s^\smallfrown(i)}$ for $i\in\{0,1\}$, $a_{s^\smallfrown(0)} \neq a_{s ^\smallfrown(1)}$. This is possible since $\tau \notin V$ but every initial segment of it is in $V$.
For every $f\in$~$^{\kappa}2$, let us pick a condition $q_f \in \mathbb{P}$ such that for all $\alpha<\kappa, q_f \leq q_{f\restriction \alpha}$ (this is possible by the closure of $\mathbb{P}$). Let $\beta_f = \sup_{\alpha < \kappa} \beta_{f\restriction \alpha}$. Then $\forall f \in$~$^{\kappa}2,$ $q_f\Vdash \tau \restriction \check\beta_f = \check{a_f}$, where $a_f = \bigcup_{\alpha < \kappa} a_{f\restriction \alpha}$, and for every $f\neq f^\prime$, $a_f \neq a_{f^\prime}$. Since we chose already only $|\alpha |<2^{\kappa}$ values for $F$, there must be some $f\in$~$^{\kappa}2$ such that $q_f \neq p_i$ for every $i < \alpha$ and $F(a_f )$ was not already defined.
At the end of this process, there might be still possible initial segment of $\tau$ such that $F$ is undefined on, so we define $F(x)$ to be arbitrary on those values.
By density arguments, in $V[G]$, $\{F(\tau \restriction \beta) : \beta < \lambda\} = (2^\kappa)^V$.
\end{proof}
We can slightly generalize the lemma, and show that every $\kappa^+$-strategically closed forcing of size $2^\kappa$ collapses a cardinal. The argument is the same, and the only difference is in the construction of the function $F$. There, in the $\alpha$-th step, when we build the tree of extensions of $p_\a,$ we use the strategy in order to ensure that in limit stages of the tree we can always find $q_s$, stronger than $q_{s\restriction \alpha}$ for every $\alpha < \dom(s)$. This means that when picking $q_{s^\smallfrown(\epsilon)}$ for $\epsilon \in \{0,1\}$ we first extend $q_s$ into two incompatible conditions that force different information about $\tau$ as we did in the former case, and then extend those conditions to $q_{s^\smallfrown(0)}, q_{s^\smallfrown(1)}$ according to the strategy, assuming that the last step was made by the bad player. The limit stages are completely defined by the strategy, since the good player plays those steps.
The lemma also holds in a slightly more general setting. For every regular cardinal $\kappa$, if $2^{<\kappa}$ is singular then any $\kappa$-closed forcing of cardinality $2^{<\kappa}$ collapses some cardinal. The proof is essentially the same.
We also need the following.
\begin{lemma}\label{lem: singular continuum everywhere}
Assume $GCH$ holds and $\kappa$ is $\aleph_{\kappa^{++}}$-strong. Then there is a generic extension of the universe in which $\kappa$ remains inaccessible and for all infinite cardinals $\lambda<\kappa, 2^{\l}$ is a singular cardinal.
\end{lemma}
\begin{proof}
We use the extender based Radin forcing as developed in \cite{merimovich}, and continued in \cite{friedman-golshani}. Our presentation follows \cite{friedman-golshani}. We assume the reader is familiar with these papers and use the definitions and results from them without any mention.
Let $V^*$ denote the ground model. Let $j\colon V^* \to M^*$ be an elementary embedding witnessing the $\aleph_{\k^{++}}$-strongness of $\k$ and let $\bar{E}\in V^*$ be an extender sequence system derived from $j$, $\bar{E}=\langle \bar{E}_\a: \a\in \dom(\bar{E}) \rangle$, where $\dom(\bar{E})=[\k, \aleph_{\k^{+}})$ and $\len(\bar{E})=\k^+.$ Then the ultrapower $j_{\bar{E}}\colon V^* \to M^*_{\bar{E}}\simeq Ult(V^*,\bar{E})$ has critical point $\k$ and $M^*_{\bar{E}}$ contains $V^*_{\aleph_{\k^{+}}}$. Consider the following elementary embeddings $\forall \tau' < \tau < \len(\bar{E})$
\begin{align*} \label{E-system}
& j_\gt\func \VS \to \MSt \simeq \Ult(\VS, E(\gt))=
\{j_\tau(f)(\bar E_\alpha \restricted \tau)\mid f\in V^*, \alpha\in [\kappa, \aleph_{\kappa^+})\},
\notag \\
& k_\gt(j_\gt(f)(\Es_\ga \restricted \gt))=
j(f)(\Es_\ga \restricted \gt),
\\
\notag & i_{\gt', \gt}(j_{\gt'}(f)(\Es_\ga \restricted \gt')) =
j_\gt(f)(\Es_\ga \restricted \gt'),
\\
\notag & \ordered{\MSE,i_{\gt, \Es}} = \limdir \ordered {
\ordof{\MSt} {\gt < \len(\Es)},
\ordof{i_{\gt',\gt}} {\gt' \leq \gt < \len(\Es)}
}.
\end{align*}
We demand that
$\Es \restricted \gt \in \MSt$ for all $\tau<\len(\bar E)$.
Thus we get the following commutative diagram.
\[
\xymatrix{
\VS \ar[rrrrr]^j \ar[rrrrd]^{j_{\Es}} \ar[rrrdd]_{j_\gt} \ar[rdd]_{j_{\gt'}} & & & & & \MS \\
& & & & \MSE\ar[ur]^{k_\Es} & \\
& M^*_{\gt'}\ar@/_6pc/[uurrrr]_{k_{\gt'}} \ar[urrr]^{i_{\gt', \Es}} \ar[rr]_{i_{\gt', \gt}} & & \MSt = \Ult(\VS, E(\gt)) \ar[ru]_{i_{\gt, \Es}}\ar@/_5pc/[rruu]_{k_\gt} & & \\
}
\]
Also factor through the normal ultrafilter to get the following commutative diagram
\begin{align*}
\begin{aligned}
\begin{diagram}
\node{\VS}
\arrow{e,t}{j_\Es}
\arrow{se,t}{j_\gt}
\arrow{s,l}{i_U}
\node{\MSE}
\\
\node{\NS \simeq \Ult(\VS, U)}
\arrow{e,b}{i_{U, \gt}}
\arrow{ne,b}{i_{U, \Es}}
\node{\MSt}
\arrow{n,b}{i_{\gt, \Es}}
\end{diagram}
\end{aligned}
\begin{aligned}
\qquad
\begin{split}
& U = E_\gk(0),
\\
& i_U \func \VS \to \NS \simeq \Ult(\VS, U),
\\
& i_{U, \gt}(i_U(f)(\gk)) = j_\gt(f)(\gk),
\\
& i_{U, \Es}(i_U(f)(\gk)) = j_\Es(f)(\gk).
\end{split}
\end{aligned}
\end{align*}
Force with
\begin{center}
$\mathbb{R}=\Add(\k^+, (\aleph_{\k^{++}})^{M^*_{\bar{E}}})\times \Add(\k^{++}, (\aleph_{\k^{+3}})^{M^*_{\bar{E}}})\times \Add(\k^{+3}, (\aleph_{j_{\bar{E}}(\k)^+})^{M^*_{\bar{E}}}).$
\end{center}
Let $G$ be $\mathbb{R}-$generic over $V$.
By essentially the same arguments as those given in \cite{friedman-golshani}, \cite{merimovich},
we can find filters $G_{\bar{E}}, G_U$ and $G_\tau, \tau<\len(\bar{E}),$ such that $G_{\bar{E}}$
is $j_{\bar{E}}(\PP)$-generic over $M^*_{\bar{E}},$ $G_U$ is $i_U(\PP)$-generic
over $N^*$ and $G_\tau$ is $j_\tau(\PP)$-generic over $M^*_\tau$
and such that
the following diagram is commutative:
\begin{align*}
\begin{diagram}
\node{V = \VS[G]}
\arrow[2]{e,t}{j_\Es}
\arrow{s,l}{i_{U}}
\arrow{se,b}{j_{\gt'}}
\arrow{see,b}{j_\gt}
\node{}
\node{\ME = \MSE[G_\Es]}
\\
\node{N = \NS[G_U]}
\arrow{e,b}{i_{U, \gt'}}
\node{M_{\gt'} = M^*_{\gt'}[G_{\gt'}]}
\arrow{ne,t,3}{i_{\gt', \Es}}
\arrow{e,b}{i_{\gt', \gt}}
\node{\Mt = \MSt[G_\gt]}
\arrow[1]{n,b}{i_{\gt, \Es}}
\end{diagram}
\end{align*}
Set
\begin{center}
$\mathbb{R}_U=\Big(\Add(\k^+, \aleph_{\k^{++}})\times \Add(\k^{++}, \aleph_{\k^{+3}})\times \Add(\k^{+3}, \aleph_{i_{U}(\k)^+})\Big)^{N^*}.$
$\mathbb{R}_\tau=\Big(\Add(\k^+, \aleph_{\k^{++}})\times \Add(\k^{++}, \aleph_{\k^{+3}})\times \Add(\k^{+3}, \aleph_{j_{\tau}(\k)^+})\Big)^{M_\tau}.$
$\mathbb{R}_{\bar{E}}=\Big(\Add(\k^+, \aleph_{\k^{++}})\times \Add(\k^{++}, \aleph_{\k^{+3}})\times \Add(\k^{+3}, \aleph_{j_{\bar{E}}(\k)^+})\Big)^{M^*_{\bar{E}}}.$
\end{center}
Then we can find $I_{U}, I_{\tau}$ and $I_{\bar{E}}$ in $V=V^*[G]$ such that:
\begin{enumerate}
\item $I_{U}$ is $\mathbb{R}_{U}$-generic over $N^{*}[G_{U}]$,
\item $I_{\tau}$ is $\mathbb{R}_{\tau}$-generic over $M_{\tau}^{*}[G_{\tau}]$,
\item $I_{\bar{E}}$ is $\mathbb{R}_{\bar{E}}$-generic over $M_{\bar{E}}^{*}[G_{\bar{E}}]$,
\item The generics are so that we have the following lifting diagram
\begin{align*}
\begin{diagram}
\node{}
\node{}
\node{\ME[I_\Es]}
\\
\node{N[I_U]}
\arrow{e,b}{i^*_{U, \gt'}}
\node{M_{\gt'}[I_{\gt'}]}
\arrow{ne,t,3}{i^*_{\gt', \Es}}
\arrow{e,b}{i^*_{\gt', \gt}}
\node{\Mt[I_\gt]}
\arrow[1]{n,b}{i^*_{\gt, \Es}}
\end{diagram}
\end{align*}
\end{enumerate}
Iterate $j_\Es$ and consider the following diagram:
\begin{align*}
\begin{diagram}
\node{V}
\arrow[2]{e,t}{j_\Es = j^{0,1}_\Es}
\arrow{se,t,1}{j_{\gt_1}}
\arrow{s,l}{i_{U}}
\node{}
\node{\ME}
\arrow[2]{e,t}{j^{1,2}_\Es}
\arrow{se,t,1}{j^2_{\gt_2}}
\arrow{s,l}{i^2_{U}}
\node{}
\node{M_\Es^2}
\arrow[2]{e,t}{j^{2,3}_\Es}
\arrow{se,t,1}{j^3_{\gt_3}}
\arrow{s,l}{i^3_{U}}
\node{}
\node{M_\Es^3}
\arrow[1]{e,..}
\\
\node{N}
\arrow[1]{e,b}{i_{U, \gt_1}}
\arrow[1]{nee,t,3}{i_{U, \Es}}
\node{M_{\gt_1}}
\arrow[1]{ne,b,1}{i_{\gt_1, \Es}}
\node{N^2}
\arrow[1]{e,b}{i^2_{U, \gt_2}}
\arrow[1]{nee,t,3}{i^2_{U, \Es}}
\node{M^{2}_{\gt_2}}
\arrow[1]{ne,b,1}{i^2_{\gt_2, \Es}}
\node{N^3}
\arrow[1]{e,b}{i^3_{U, \gt_3}}
\arrow[1]{nee,t,3}{i^3_{U, \Es}}
\node{M^{3}_{\gt_3}}
\arrow[1]{ne,b,1}{i^3_{\gt_3, \Es}}
\end{diagram}
\end{align*}
where
\begin{align*}
& j^0_\Es = \id,
\\
& j^{n}_\Es = j^{0, n}_\Es,
\\
& j^{m, n}_\Es = j^{n-1, n}_\Es \circ \dotsb \circ j^{m+1, m+2}_\Es \circ j^{m ,m+1}_\Es.
\end{align*}
Let $R(-,-)$ be a function such that
\begin{center}
$i_{U}^{2}(R)(\kappa, i_{U}(\kappa))=\mathbb{R}_U,$
\end{center}
where $i_{U}^{2}$ is the second iterate of $i_{U}.$ It is also clear that
\begin{center}
$j_{\bar{E}}^{2}(R)(\kappa, j_{\bar{E}}(\kappa))=\mathbb{R}_{\bar{E}}.$
\end{center}
As in \cite{merimovich} (see also \cite{friedman-golshani}), in the prepared model $V=V^*[G]$, we define a new extender sequence system $\bar{F}= \langle \bar{F}_{\alpha}: \alpha \in \dom(\bar{F})\rangle$ by:
\begin{itemize}
\item $\dom(\bar{F})=\dom(\bar{E}),$ \item $\len(\bar{F})=\len(\bar{E})$ \item $\leq_{\bar{F}}=\leq_{\bar{E}},$ \item $F(0)=E(0),$ \item $I(\tau)=I_{\tau},$ \item $\forall 0< \tau < \len(\bar{F}), F(\tau)= \langle \langle F_{\alpha}(\tau): \alpha \in \dom(\bar{F}) \rangle, \langle \pi_{\beta, \alpha}: \beta, \alpha \in \dom(\bar{F}), \beta \geq_{\bar{F}} \alpha \rangle \rangle$ is such that
\begin{center}
$X \in F_{\alpha}(\tau) \Leftrightarrow \langle \alpha, F(0), I(0), ..., F(\tau^{'}), I(\tau^{'}), ...: \tau^{'} < \tau \rangle \in j_{\bar{E}}(X),$
\end{center}
and
\begin{center}
$\pi_{\beta, \alpha}(\langle \xi, d \rangle)= \langle \pi_{\beta, \alpha}(\xi), d \rangle, $
\end{center}
\item $\forall \alpha \in \dom(\bar{F}), \bar{F}_{\alpha}= \langle \alpha, F(\tau),I(\tau): \tau < \len(\bar{F}) \rangle.$
\end{itemize}
Also let $I(\bar{F})$ be the filter generated by $\bigcup_{\tau < \len(\bar{F})} i_{\tau, \bar{E}}^{''}I(\tau).$ Then $I(\bar{F})$ is $\mathbb{R}_{\bar{F}}$-generic over $M_{\bar{F}}.$
Working in $V$, let $\mathbb{P}_{\bar{F}}$ be the corresponding extender based Radin forcing, as defined in \cite{merimovich} (see also \cite[Definition 5.1]{friedman-golshani}). Note that the definition of $\mathbb{P}_{\bar{F}}$ depends on the function $R$, and this is were the iterability of the extender sequence plays a role. Let $H$ be $\mathbb{P}_{\bar{F}}$-generic over $V$. By reflection, we may assume that each $\bar{\mu}$ which appears in some condition in $\mathbb{P}_{\bar{F}}$ has $\dom(\bar{\mu})=[\k^0(\bar{\mu}), \aleph_{\k^0(\bar{\mu})^{+}}).$ For $\alpha \in \dom(\bar{F})$ set
\begin{center}
$C_{H}^{\alpha} = \{ \max\kappa(p_{0}^{\bar{F}_{\alpha}}): p \in H \}.$
\end{center}
The following clauses can be proved as in \cite{friedman-golshani}, \cite{merimovich}.
\begin{enumerate}
\item $V[H]$ and $V$ have the same cardinals.
\item $\kappa$ remains strongly inaccessible in $V[H]$.
\item $C_{H}^{\kappa}$ is a club in $\kappa.$
\item Let $\k_0=\min(C_{H}^{\kappa}),$ and let $K$ be $\Add(\omega, \aleph_{\k_0^{+}})_{V[H]}$-generic over $V[H].$ Then the following hold in $V[H][K]:$
\begin{itemize}
\item $\forall \l <\k_0, 2^\l=\aleph_{\k_0^+}$.
\item If $\l < \l_*$ are two successive points in $C_{H}^{\kappa}$, then $2^{\l^{+n}}=\aleph_{\l^{+n+1}},$ for $n=0,1,2$ and $2^{\l^{+3}}=\aleph_{\l_*^+}$.
\end{itemize}
\end{enumerate}
In particular $V[H][K] \models $``$\k$ is strongly inaccessible and for all $\l<\k,$ $2^\l$ is a singular cardinal''. The lemma follows.
\end{proof}
We are now ready to complete the proof of Theorem 3.1.
Let $V$ be a model of $GCH+\k$ is $\aleph_{\kappa^{++}}$ strong. By Lemma~\ref{lem: singular continuum everywhere}, there exists a cardinal preserving generic extension $V$ of $V^*$ in which $\k$ remains inaccessible, and for all infinite cardinals $\l<\k, 2^\l$ is a singular cardinal. Let $\k_*\leq \k$ be the least inaccessible cardinal of $V$, and consider the model $V_{\k_*}.$ It is a model of $ZFC$. We show that $V_{\k_*}$ is as required. So let $\l<\k_*$ be an uncountable cardinal, and let $\mathbb{P}\in V_{\k_*}$ be a non-trivial $\l$-closed forcing notion of size $2^{<\l}.$ As for a singular cardinal $\l,$ being $\l$-closed implies $\l^+$-closed, we can assume without loss of generality that $\l$ is a regular cardinal, and hence by our choice of $\k_*, \l=\mu^+,$ for some cardinal $\mu.$ So it follows from Lemma~\ref{lem: singular continuum collapsing} that forcing with $\mathbb{P}$ collapses $2^\mu$. The theorem follows.
\end{proof}
\section{consistency strength of statement $(2)$ for forcing notions of arbitrary size}
In this section we discuss the consistency of statement $(2)$ for forcing notions of arbitrary large size. We show that in this case the problem is much more difficult, and it requires some very large cardinals. For the sake of simplicity, let's just consider the case $\kappa=\aleph_0.$
The following shows that non-triviality of the forcing is essential. Suppose $\lambda>\aleph_1$ is regular and let $\mathbb{P}=\lambda$, as set of ordinals. Order $\mathbb{P}$ with the reversed ordinal order separated at $\omega_1$, i.e. $p \leq_\mathbb{P} q$ iff $p \geq q$ as ordinals and $p,q < \omega_1$ or $\omega_1 \leq p,q$.
It is clear that $\mathbb{P}$ is $\aleph_1$-closed (but not $\aleph_2$-closed), and it preserves all cardinals.
\begin{theorem}
Let $\l$ be a regular cardinal such that $2^{<\l} = \l$. For every regular cardinal $\mu < \l$ there is a separative forcing notion $\mathbb{P}$ which is $\mu$-closed, not $\mu^+$-closed and $\l$-distributive. Moreover, $|\mathbb{P}| = \l$, so $\mathbb{P}$ does not collapse cardinals.
\end{theorem}
\begin{proof}
The forcing $\mathbb{P}$ will be the forcing that adds a non-reflecting stationary subset of $S^{\l}_{<\mu} = \{\alpha < \l \mid \cf\alpha < \mu\}$, by bounded conditions. Let us describe $\mathbb{P}$ precisely:
The conditions in $\mathbb{P}$ are functions $p \colon S^{\l}_{<\mu} \cap \alpha \to 2$ such that $\alpha < \l$ and $p^{-1}(1)$ is non-stationary at every $\beta \leq \alpha$ such that $\cf\beta\geq\mu$.
The order of $\mathbb{P}$ is end extension. Let us show that $\mathbb{P}$ has the required properties.
First, since $2^{<\l} = \l$, there are exactly $\l$ bounded subsets of $\l$. In particular, $|\mathbb{P}| \leq \l$. On the other hand, for every $\alpha \in S^{\l}_{<\mu}$, the function $p\colon S^{\l}_\mu \cap \alpha \to 2$ defined by $p(\b) = 0$ for every $\b$ is a condition in $\mathbb{P}$, so $|\mathbb{P}| = \l$.
Next, $\mathbb{P}$ is $\mu$-closed since if $\langle p_\alpha \mid \alpha < \eta\rangle$, $\eta < \mu$, is a decreasing sequence of conditions, then $p = \bigcup_{\alpha<\eta} p_\alpha$ is a condition in $\mathbb{P}$. The only thing that we need to verify is that $p^{-1}(1)$ is non-stationary, or that $\cf(\dom(p)) < \mu$, and clearly if the sequence is not eventually constant the second option occurs.
Let us show that $\mathbb{P}$ is not $\mu^+$-closed. For each $\alpha< \mu$ let $p_\alpha \in \mathbb{P}$ be the condition with $\dom(p_\alpha) = \alpha$ and $p_\alpha (\b) = 1$ for every $\b$. Then the sequence of all $p_\alpha$, $\alpha < \mu$ has no lower bound, since for any condition $q$ such that $q\leq p_\alpha$ for every $\alpha$ we would have $\alpha \subseteq q^{-1}(1)$ for every $\alpha < \mu$ and therefore $\mu \subseteq q^{-1} (1)$, so $q^{-1}(1)$ is stationary at $\mu$. Similar argument shows that we can construct such sequences below any condition $p \in \mathbb{P}$, so $\mathbb{P}$ is nowhere $\mu^+$-closed.
The main property of $\mathbb{P}$ is its $\l$-distributiveness. Let us show that $\mathbb{P}$ is $\l$-strategically closed, and in particular $\l$-distributive.
Let $\langle p_\alpha \mid \alpha < \eta\rangle$ be the play until step $\eta$. If $\eta$ is a successor ordinal, the strategy of the good player is to pick some ordinal in $S^{\l}_{<\mu}$ above the supremum of the domain of $p_{\eta - 1}$ and extend $p_{\eta - 1}$ up to this ordinal by appending zeros. If $\eta$ is limit less than $\l$ and $\bigcup_{\alpha < \eta} p_\alpha$ is a condition (so the good player did not lose already), the strategy will be to set $p_\eta = \bigcup_{\alpha < \eta} p_\alpha \cup \{\langle\eta,0 \rangle\}$, if $\eta \in S^{\l}_{<\mu}$, and otherwise just $p_\eta = \bigcup_{\alpha < \eta} p_\alpha$.
Let us prove that this is indeed a winning strategy, namely that at every limit stage of the play below $\l$, the union of the conditions until this step is a condition. Let $\eta$ be a limit ordinal of cofinality at least $\mu$. Then $\{\supp(p_\alpha) : \alpha < \eta,\,\text{limit}\}$ is a club at $\sup_{\alpha < \eta}\supp (p_\alpha)$ that witnesses that $\bigcup p_\alpha ^{-1} (1)$ is non-stationary, as wanted.
Since $\mathbb{P}$ is $\l$-distributive, it doesn't collapse cardinals $\leq \l$. Moreover, since $|\mathbb{P}|=\l$, $\mathbb{P}$ is $\l^{+}$-$c.c.$ and therefore it does not collapse cardinals above $\l$, so $\mathbb{P}$ does not collapse cardinals at all.
\end{proof}
\begin{corollary}
Assume there is an uncountable regular cardinal $\l$ such that $2^{<\l}=\l.$ Then there is a non-trivial $\aleph_1$-closed (but not $\aleph_2$-closed) forcing notion which preserves all cardinals.
\end{corollary}
\begin{corollary}
Suppose there is no inner model with a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}.$ Then there is a non-trivial $\aleph_1$-closed but not $\aleph_2$-closed forcing notion which preserves all cardinals.
\end{corollary}
\begin{proof}
By results of Gitik and Mitchell \cite{mitchell}, there is a strong limit singular cardinal $\kappa$ such that $2^\kappa=\kappa^+$. Applying the previous lemma with $\mu=\aleph_1$ we get the desired result.
\end{proof}
It follows that if we want a model in which all non-trivial $\aleph_1$-closed forcing notions collapse some cardinals, then $GCH$ should fail everywhere in that model.
Let us close this section by showing that there exists a non-trivial $\aleph_1$-closed cardinal preserving forcing notion in the model $V[H][K]$ of section 3. We assume that $V^*$ (and hence $V=V^*[G]$) has no inner model with a Woodin cardinal. It then follows that the combinatoric principle $\Box_\mu$ holds for every singular $\mu$ (see \cite{schimmerling-zeman}).
Let $\mu=\aleph_{\k_0^+}$ and $\l=\mu^+.$ Note that $V[H][K]=V[K][H],$ and that
\begin{center}
$V[K]\models$``$2^{\aleph_0}=\mu + \Box_\mu + SCH$ holds at $\mu$''.
\end{center}
It follows from \cite{rinot} that there exists an $\aleph_1$-closed $\l$-Souslin tree $T$ in $V[K]$. We show that $T$ remains an $\aleph_1$-closed $\l$-Souslin tree in $V[K][H].$
To do this, we need a finer analysis of the model $V[K][H].$ For inaccessible cardinals $\a<\b$ set
\begin{center}
$\mathbb{P}(\a,\b)=(\Add(\a^+, \aleph_{\a^{++}})\times \Add(\a^{++}, \aleph_{\a^{+3}})\times Add(\a^{+3}, \aleph_{\b^+}))^V.$
\end{center}
Then it is easily seen that
\begin{center}
$V[K][H]=V[K\times H_1][H_2]=V[K][H_1][H_2],$
\end{center}
where
\begin{enumerate}
\item $K\times H_1$ is $\Add(\aleph_0, \mu)\times\mathbb{P}(\k_0, \k_1)$-generic over $V,$ where $\k_0 < \k_1$ are the first two elements of the club $C_H^\k.$
\item $V_{\k_1}^{V[K][H]}=V_{\k_1}^{V[K][H_1]}.$
\end{enumerate}
Now $V\models$``$\Add(\aleph_0, \mu)\times\mathbb{P}(\k_0, \k_1)$ is $\k_0^{+4}$-Knaster'', hence $V[K]\models$``$\mathbb{P}(\k_0, \k_1)$ is $\k_0^{+4}$-Knaster, and $\k_0^+$-distributive''.
So after forcing with $\mathbb{P}(\k_0, \k_1)$ over $V[K]$, $T$ remains an $\aleph_1$-closed $\l$-Souslin tree. So by clause $(2)$ above
\begin{center}
$V[K][H]\models$`` $T$ is an $\aleph_1$-closed $\l$-Souslin tree''.
\end{center}
Thus $T$, considered as a forcing notion, is a non-trivial $\aleph_1$-closed forcing notion of size $\l$ which preserves all cardinals.
\section{Adding club sets in the absence of $GCH$}
In this section, we continue our study of the last section, and present a different method for producing cardinal preserving forcing notions in the absence of instances of $GCH$. Our forcing notions will be of size of a regular cardinal $\l$ such that $\l^{<\l} = \l$. Given any regular cardinal $\kappa$ with $\kappa^+ < \lambda,$ note that there is a natural forcing (i.e., $\Add(\l,1)$) which is $\kappa^+$-closed and $\l$-distributive. However the forcing $\Add(\l, 1)$
is also $\kappa^{++}$-closed and it preserves stationary subsets of $\lambda.$ The next theorem shows that it is consistent that there are
$\kappa^+$-closed but not $\kappa^{++}$-closed forcing notions of size $\lambda$ which are $\l$-distributive and destroy a stationary subset of $\lambda$.
The results of this section are joint work with Moti Gitik, and are presented here with his kind permission.
\begin{theorem} Assume $GCH$. Let $\kappa$ and $\lambda>\kappa^+$ be regular cardinals. Then there is a cofinality preserving generic extension in which the following hold: $2^\kappa=\lambda$ and there exists a non-trivial $\kappa^+$-closed, but not $\kappa^{++}$-closed, forcing notion of size $\lambda$ which preserves all cardinals (in fact the forcing is $\lambda$-distributive, $\lambda^+$-$c.c.$).
\end{theorem}
\begin{proof}
Let $\mathbb{P}_1=Add(\lambda, 1)$ and let $G_1$ be $\mathbb{P}_1$-generic over $V$. Let $F=\bigcup G_1.$ Then $F: \lambda \rightarrow 2.$ Let $\mathbb{P}_2=Add(\kappa, \lambda)^{V[G_1]}$ and let $G_2$ be $\mathbb{P}_2$-generic over $V[G_1].$ In $V[G_1*G_2], 2^\kappa=\lambda.$ We show that in $V[G_1*G_2],$ there is a non-trivial $\kappa^+$-closed, but not $\kappa^{++}$-closed, forcing notion of size $\lambda$ which preserves all cardinals. Let
\begin{center}
$S=\{ \alpha <\lambda : cf(\alpha)\leq \kappa$ or $(cf(\alpha)>\kappa, F(\alpha)=1) \}.$
\end{center}
$S$ is easily seen to be a stationary co-stationary subset of $\lambda$ in models $V[G_1]$ and $V[G_1*G_2]$. Let $\mathbb{R} \in V[G_1*G_2]$ be the forcing notion for adding a club into $S$ by approximations of cardinality $<\lambda.$
\begin{claim}
$(a)$ $\mathbb{R}$ is $\lambda^+$-$c.c.$
$(b)$ $\mathbb{R}$ is $\kappa^+$-closed but not $\kappa^{++}$-closed.
$(c)$ $\mathbb{R}$ is $\lambda$-distributive.
\end{claim}
\begin{proof}
$(a)$ and $(b)$ are trivial, so let's prove $(c).$ Let $K$ be $\mathbb{R}$-generic over $V[G_1*G_2]$,
and let $h\in V[G_1*G_2*K], h:\mu \rightarrow On,$ where $\mu<\l$ is regular. We show that $h\in V[G_1*G_2].$
Work in $V[G_1 * G_2]$. Let $F=\bigcup G_1: \lambda \rightarrow 2$ be as above and let $\theta$ be large enough regular. By density arguments, we can find a continuous chain $\langle M_\a: \a<\mu \rangle$ of elementary submodels of $H(\theta)$ such that:
\begin{enumerate}
\item Every initial segment of $\langle M_\a: \a<\mu \rangle$ is in $V$,
\item $|M_\a| <\l,$
\item $\a<\b \Rightarrow M_\a \subseteq M_\beta,$
\item $\langle M_\a: \a<\beta \rangle\in M_{\beta+1},$
\item $\forall \a<\mu, M_\a\cap \l\in \l,$
\item $\forall \a<\mu, F(M_\a\cap \l)=1,$
\item $F(M\cap \l)=1,$ where $M=\bigcup_{\a<\mu}M_\a.$
\end{enumerate}
We may define the sequence $\langle M_\a: \a<\mu \rangle$ in such a way that
$M$ contains all the relevant information. Let $\delta=M\cap\l$. Using the $\lambda$-closure of $Add(\lambda,1)$ and the chain condition of $Add(\kappa,\lambda)$, we can easily arrange $G_1\upharpoonright \delta=G_1\cap \Add(\delta, 1)^M$ is $\Add(\delta, 1)^M$-generic over $M$ and $\{\a<\delta: F(\a)=1 \}$ contains a club $C$ in $\delta$ (e.g. $C = \{ M_\a \cap \l \mid \alpha < \delta\}$). By our construction we may also assume that all the initial segments of $C$ are in $M[G_1\upharpoonright\delta].$ Let $G_2\upharpoonright \delta=G_2\cap \Add(\k,\delta)^{M[G_1\upharpoonright\delta]}.$ Then $G_2\upharpoonright\delta$ is
$\Add(\k,\delta)^{M[G_1\upharpoonright\delta]}$-generic over $M[G_1\upharpoonright\delta].$
Consider the model $N=M[G_1\upharpoonright\delta *G_2\upharpoonright\delta].$ Now we can decide values of $h$ inside $N$ and use $C$ to ensure that the conditions of $\mathbb{Q}$
used in the process go up to $\delta$. This will allow us to extend them finally to a single
condition deciding all the values of $h$. It follows that every initial segment of $h$ is in $N$ and hence $h\in V[G_1*G_2],$ as required.
\end{proof}
It follows that $\mathbb{R}$ preserves all the cardinals, and the theorem follows.
\end{proof}
Although we shoot a club through the fat stationary set $S = \{ \alpha < \kappa \mid F(\alpha) = 1\}$, we can't use Abraham-Shelah's general theorem from \cite{abraham-shelah}, since their cardinal arithmetic assumptions do not hold.
We now present a generalization of the above theorem, whose proof is essentially the same.
\begin{theorem}
Let $\k$ and $\l>\k^+$ be regular cardinals, and let $V[G_1*G_2]$ be a generic extension of $V$ by $\Add(\l,1)*\lusim{\Add}(\k, \l).$ Suppose $\mathbb{Q}\in V[G_1*G_2]$ is a cardinal preserving forcing notion of size $<\l,$ and let $H$ be $\mathbb{Q}$-generic over $V[G_1*G_2]$. Let $S=\{ \alpha <\lambda : cf(\alpha)\leq \kappa$ or $(cf(\alpha)>\kappa, \bigcup G_1(\alpha)=1) \}$. Then $S$ is a stationary subset of $\l$ in $V[G_1*G_2*H],$ and if $\mathbb{R}$ is the forcing notion for adding a club subset of $S$, using approximations of size $<\l,$ then
\begin{center}
$V[G_1*G_2*H]\models$``$\mathbb{R}$ is a $\k^+$-closed cardinal preserving forcing notion''.
\end{center}
\end{theorem}
As an application of the above theorem, let us prove an analogue of Theorem 5.1 for $\k$ singular.
\begin{theorem}
Suppose $\kappa$ is a strong cardinal and $\lambda>\kappa^+$ is regular. Then there is a forcing extension in which the following hold: $\kappa$ is a strong limit singular cardinal, $2^\kappa=\lambda$ and there exists a non-trivial $\kappa^+$-closed, but not $\kappa^{++}$-closed, cardinal preserving forcing notion of size $\lambda$.
\end{theorem}
\begin{proof}
By results of Gitik-Shelah and Woodin \cite{gitik-shelah}, we can assume that $\k$ is indestructible under the following kind of forcing notions:
\begin{itemize}
\item $\Add(\k, \delta),$ for any ordinal $\delta,$
\item Any $\k^+$-weakly closed forcing notion which satisfies the Prikry condition.
\end{itemize}
Fix any regular cardinal $\delta<\k.$ Force with $\Add(\l,1)*\lusim{\Add}(\k, \l)$ and let $G_1*G_2$ be $\Add(\l,1)*\lusim{\Add}(\k, \l)$-generic over $V$. Then $\k$ remains strong in $V[G_1*G_2].$ Let $\mathbb{Q}$ be the Prikry or Magidor forcing for changing the cofinality of $\k$ into $\delta$ and let $H$ be $\mathbb{Q}$-generic over $V[G_1*G_2].$ Then by Theorem 5.3, we can find, in $V[G_1*G_2*H],$ a $\k^+$-closed, but not $\k^{++}$-closed cardinal preserving forcing notion.
\end{proof}
\section{Consistently, for all $\k, 2^\kappa>\k^+$ and there is a $\kappa^+$-closed cardinal preserving forcing notion}
In this section, we again consider statement $(2),$ and prove a global consistency result, which is, in some sense, in the opposite direction of Theorem 3.1.
\begin{theorem}
Assuming the existence of a strong cardinal and infinitely many inaccessible cardinals above it, it is consistent that $GCH$ fails everywhere, and for each regular uncountable cardinal $\k,$ there exists a non-trivial $\k$-closed forcing notion which preserves all cardinals.
\end{theorem}
\begin{proof}
To prove the theorem, we need two auxiliary lemmas.
\begin{lemma}
Assume $2^\k$ is weakly inaccessible, and for all $\k<\l<2^\k, 2^\l=2^\k.$ Then there exists a $\k^+$-closed forcing notion which preserves all cardinals.
\end{lemma}
\begin{proof}
By Lemma 4.1.
\end{proof}
The next lemma is a generalization of the main theorem of Foreman-Woodin \cite{foreman-woodin}, where the use of a supercompact cardinal with infinitely many inaccessible
cardinals above it, is replaced by the much weaker assumption of the existence of a strong cardinal with an inaccessible above it.
\begin{lemma}
Assume $GCH$ holds, $\k$ is a strong cardinal and there exists an inaccessible cardinal above $\k$. Then there is a generic extension in which $\k$ remains inaccessible, for all $\l<\k, 2^\l$ is weakly inaccessible and $\l<\mu<2^\l$ implies $2^\mu=2^\l.$
\end{lemma}
\begin{proof}
The proof is similar to the proof of Lemma 3.4, so we follow that proof and just mention the changes which are required.
Let $V^*$ denote the ground model.
Also, for an ordinal $\alpha$, let us denote the next inaccessible cardinal above $\alpha$ by $\inacc(\alpha)$.
Let $j: V^* \rightarrow M^*$ be an elementary embedding witnessing the $\inacc(\kappa)^+$-strongness of $\kappa$ and let $\bar{E} \in V^*$
be an extender sequence system derived from $j$, $\bar{E}=\langle \bar{E}_\a: \a\in \dom(\bar{E}) \rangle$, where $\dom(\bar{E})=[\k, \inacc(\kappa))$ and $\len(\bar{E})=\k^+.$ Then the ultrapower $j_{\bar{E}}\colon V^* \to M^*_{\bar{E}}\simeq Ult(V^*,\bar{E})$ has critical point $\k$ and $M^*_{\bar{E}}$ contains $V^*_{\inacc(\kappa)}$. As before, consider the resulting elementary embeddings $j_{\bar{E}}, j_\tau, k_\tau, i_{\tau', \tau}$ and $i_{\tau, \bar{E}}$, where
$\tau' < \tau < \kappa^+.$
Also factor through the normal ultrafilter to get the normal measure $U$ and embeddings $i_U, i_{U, \tau}$
and $i_{U, \bar{E}}$.
Force with
\begin{center}
$\MPB=\Add(\inacc(\kappa), \inacc(\kappa)^{+3})$.
\end{center}
Let $G$ be $\MPB$-generic over $V^*$ and let $V=V^*[G]$. As before, we can find suitable
filters $G_{\bar{E}}, G_U$ and $G_\tau, \tau<\len(\bar{E}),$ such that $G_{\bar{E}}$
is $j_{\bar{E}}(\PP)$-generic over $M^*_{\bar{E}},$ $G_U$ is $i_U(\PP)$-generic
over $N^*$ and $G_\tau$ is $j_\tau(\PP)$-generic over $M^*_\tau$ and such that the resulting
diagram commutes.
Set
$\MPB_U= \Add(\inacc(\kappa), i_U(\kappa))^{N^*},$
and define the forcing notions $\MPB_\tau$
and $\MPB_{\bar{E}}$ similarly, where $i_U, N^*$ are replaced with
$j_\tau, M^*_\tau$ and $j_{\bar{E}}, M^*_{\bar{E}}$ respectively.
We show that there are
$I_{U}, I_{\tau}$ and $I_{\bar{E}}$ in $V=V^*[G]$ such that
$I_{U}$ is $\mathbb{P}_{U}$-generic over $N^{*}[G_{U}]$,
$I_{\tau}$ is $\mathbb{P}_{\tau}$-generic over $M_{\tau}^{*}[G_{\tau}]$, $I_{\bar{E}}$ is $\mathbb{P}_{\bar{E}}$-generic over $M_{\bar{E}}^{*}[G_{\bar{E}}]$, and the generics are so that the corresponding diagram lifts.
Set $I_{\bar{E}}=G \cap \MPB_{\bar{E}}.$ We show that it is $\MPB_{\bar{E}}$-generic over $M_{\bar{E}}^{*}[G_{\bar{E}}]$. Using Easton's lemma, it suffices to
show genericity over $M_{\bar{E}}^{*}.$ Let $A \subseteq \mathbb{P}_{\bar{E}}$ be a maximal antichain in $M_{\bar{E}}^{*}$
and $X= \bigcup \{\dom(p): p \in A \}.$ Then $|X| \leq \inacc(\kappa),$ and $A$ is a maximal antichain of $\Add(\inacc(\kappa), X)^{M_{\bar{E}}^{*}}$.
As $\Add(\inacc(\kappa), X)^{M_{\bar{E}}^{*}}=\Add(\inacc(\kappa), X)^{V^*}$, $A$ is a maximal antichain of $\Add(\inacc(\kappa), X)^{V^*},$ hence a maximal antichain
of $\Add(\inacc(\kappa), \inacc(\kappa)^{+3})^{V^*}$. Let $p \in A \cap G(\kappa).$ Then $p \in A \cap I_{\bar{E}}.$
Now set
\begin{center}
$I_U = \langle i_{U, \bar{E}}^{-1''}[I_{\bar{E}}] \rangle,$ ~the filter generated by ~ $i_{U, \bar{E}}^{-1''}[I_{\bar{E}}]$
\end{center}
and
\begin{center}
$I_\tau = \langle i_{\tau, \bar{E}}^{-1''}[I_{\bar{E}}] \rangle,$ ~the filter generated by ~ $i_{\tau, \bar{E}}^{-1''}[I_{\bar{E}}]$
\end{center}
It is easily seen that $I_U$ and $I_\tau$ are respectively $\MPB_U$-generic over $N^*$ and $\MPB_\tau$-generic over $M^*_\tau$. Now by applying Easton's lemma, we can conclude the desired result.
Let $R(-,-)$ be a function such that
\begin{center}
$i_{U}^{2}(\kappa, i_{U}(\kappa))=\mathbb{P}_U,$
\end{center}
where $i_{U}^{2}$ is the second iterate of $i_{U}.$ Working in $V=V^*[G],$ define the new extender sequence system
$\bar{F}= \langle \bar{F}_{\alpha}: \alpha \in \dom(\bar{F})\rangle$ as before.
Working in $V$, let $\mathbb{P}_{\bar{F}}$ be the corresponding extender based Radin forcing, and let $H$ be $\mathbb{P}_{\bar{F}}$-generic over $V$. By reflection, we may assume that each $\bar{\mu}$ which appears in some condition in $\mathbb{P}_{\bar{F}}$ has $\dom(\bar{\mu})=[\k^0(\bar{\mu}), \inacc(\k^0(\bar{\mu}))).$ For $\alpha \in \dom(\bar{F})$ set
\begin{center}
$C_{H}^{\alpha} = \{ \max\kappa(p_{0}^{\bar{F}_{\alpha}}): p \in H \}.$
\end{center}
The following clauses can be proved as before:
\begin{enumerate}
\item $V[H]$ and $V$ have the same cardinals.
\item $\kappa$ remains strongly inaccessible in $V[H]$
\item $C_{H}^{\kappa}$ is a club in $\kappa,$
\item If $\l < \l_* $ are two successive points in $C_{H}^{\kappa}$, then $2^\l = \inacc(\lambda)$
and $2^{\inacc(\l)}=\l_*,$
\item Let $\gamma_0=\min(C_{H}^{\kappa}).$ If $\gamma_0\leq \l< \mu<2^\l<\k$, then $2^\mu=2^\l.$
\end{enumerate}
It follows immediately that
\begin{center}
$V[H]\models$`` If $\gamma_0\leq \l<\k,$ then $2^\l$ is weakly inaccessible and $\l< \mu<2^\l \Rightarrow 2^\mu=2^\l$''.
\end{center}
Force over $V[H]$ with $Add(\aleph_0, \gamma_0),$ and let $K$ be $Add(\aleph_0, \gamma_0)$-generic over $V[H].$ Then
\begin{center}
$V[H][K]\models$``If $\l<\k,$ then $2^\l$ is weakly inaccessible and $\l< \mu<2^\l \Rightarrow 2^\mu=2^\l.$
\end{center}
The lemma follows.
\end{proof}
We are now ready to complete the proof of Theorem 6.1.
Let $V^*$ be a model of $GCH+\k$ is a strong cardinal $+$ there are infinitely many inaccessible cardinals above $\k.$ By Lemma 6.3 there exists a generic extension $V$ of $V^*$ in which $\k$ remains measurable and for all infinite cardinals $\l<\k, 2^\l$ is weakly inaccessible and if $\l<\mu<2^\l,$ then $2^\mu=2^\l.$ Let $\k_*\leq \k$ be the least inaccessible of $V$, and consider the $ZFC$ model $V_{\k_*}.$ By Lemma 6.2, $V_{\k_*}$ is as required and the theorem follows.
\end{proof}
\begin{remark}
The use of an inaccessible cardinal above the strong cardinal is essentially to help us in constructing a generic extension in which the power function is such that for all infinite cardinals $\kappa, 2^\kappa$ is weakly inaccessible and if $\kappa <\lambda<2^\kappa$ then $2^\lambda=2^\kappa.$ It seems that we need such a behavior of the power function if we want to produce cardinal preserving forcing notions using Theorem 4.1, as otherwise the corresponding forcing will have a larger size and we will face some troubles in checking the chain condition.
\end{remark}
Note that in the above arguments, we always have $\k<\l<2^\k \Rightarrow 2^\l=2^\k,$ so it is natural to ask if it is consistent that $\k^+ < 2^\k < 2^{<2^\k}$ and there exists a $\k^+$-closed
( but not $\k^{++}$-closed) forcing notion which preserves all cardinal.
The next lemma gives a positive answer to this question.
\begin{lemma}
Suppose $GCH$ holds and $\k < \mu < \delta < \l$ are regular cardinals. Let $\mathbb{P}=\Add(\k, \delta),$ $\mathbb{Q}=\Add(\mu, \l)$ and let $G\times H$ be $\mathbb{P}\times \mathbb{Q}$-generic over $V$. Let $\mathbb{R}=\Add(\delta, 1)_{V[G]}.$ Then the following hold in $V[G \times H]:$
$(a)$ $\mathbb{R}$ is $\mu$-closed but not $\mu^+$-closed.
$(b)$ $\mathbb{R}$ is $\delta$-distributive and has size $\delta$.
In particular $V[G\times H]\models$`` $2^\k=\delta, 2^\mu=\l$ and there exists a $\mu$-closed (but not $\mu^+$-closed ) forcing notion of size $\delta=2^{<\mu}$ which preserves all cardinals''.
\end{lemma}
\begin{remark}
In fact our proof shows that it suffices to have:
\begin{itemize}
\item $\mathbb{P}$ is $\delta$-Knaster and it forces $2^{<\delta}=\delta,$
\item $\mathbb{Q}$ is $\mu$-closed and $\delta$-$c.c.$
\end{itemize}
\end{remark}
\begin{proof}
Recall that since $\mathbb{P}$ is $\kappa^+$-$c.c.$ and $\mathbb{Q}$ is $\mu$-closed, $\mu \geq \kappa^+$, by Easton's lemma,
\begin{center}
$\Vdash_{\mathbb{P} } \mathbb{Q}$ is
$\mu$-distributive, and
$\Vdash_{\mathbb{Q} } \mathbb{P}$ is $\kappa^+$-c.c.
\end{center}
Note that $V[G]\models \mathbb{R}$ is $\delta$-closed, and in particular $\mu$-closed. $\mathbb{Q}$ does not add any sequence of elements of $\mathbb{R}$ of length shorter than $\mu$, so $\mathbb{R}$ is still $\mu$-closed in $V[G][H]$.
Let us show now that $\mathbb{R}$ is $\delta$-distributive in $V[G][H]$ (and in particular it does not collapse cardinals).
For this end, we first show that $\mathbb{Q}$ is $\delta$-$c.c.$ after forcing with $\mathbb{P}\ast \lusim{\mathbb{R}}$. Assume otherwise, and let $\mathcal{A}\subseteq \mathbb{Q}$ be an antichain of cardinality $\delta$. Since $\mathbb{R}$ is $\delta$-closed in $V[G]$, one can decide the values of $\mathcal{A}$ by induction on $\alpha < \delta$ and obtain an antichain of cardinality $\delta$ in $V[G]$.
Let $\langle a_i \mid i < \delta \rangle$ enumerates $\mathcal{A}$ in $V[G]$ and let $p_i \Vdash$`` $\lusim{a}_i = \check{q_i}$'' by some condition $p_i$ in $\mathbb{P}$, where $\lusim{a}_i$ is a $\mathbb{P}$-name for $a_i$. Since $\mathbb{P}$ is $\delta$-Knaster there is $I\subset \delta$ such that for every $i, j \in I$, $p_i$ is compatible with $p_j$. Therefore, $q_i \perp q_j$ for every $i \neq j$ in $I$, and this implies that already in $V$ there is an antichain in $\mathbb{Q}$ of cardinality $\delta$, which is a contradiction.
Let $K$ be a $V[G][H]$-generic for $\mathbb{R}$ and let $x$ be a sequence of ordinals of length $<\delta$ in $V[G][H][K]$. Since $V[G][H][K] = V[G][K][H]$, and $\mathbb{Q}$ is $\delta$-$c.c.$ in $V[G][K]$, there is a set $\lusim{x}$ of cardinality $<\delta$ which is a $\mathbb{Q}$-name for $x$ (so $\lusim{x} \subseteq \mathbb{Q}\times On$).
Since $\mathbb{R}$ is $\delta$-distributive in $V[G]$, $\lusim{x}\in V[G]$. Therefore, $x\in V[G][H]$, as wanted.
\end{proof} | 127,727 |
TITLE: All-pairs shortest paths in trees?
QUESTION [12 upvotes]: This is a reference request, since I'm sure what follows isn't new, but I can't seem to find it.
Suppose that we have a finite tree $T$ with non-negative weights on the edges. Naively, computing the path lengths (i.e., sum of the weights along the unique path) between every pair requires $O(n^3)$ steps: there are $\binom{n}{2}$ pairs of vertices and we can always bound the number of edges on any path by $n-1$.
We can, however, do a great deal better with the following trick. Pick a root $r$ for $T$ arbitrarily. Define the least common ancestor of $i$ and $j$ as the vertex $a$ where the path from $i$ to $r$ meets the path from $j$ to $r$. Then if $d(\cdot,\cdot)$ denotes the distance in $T$, we get $d(i,j) = d(i,r) + d(j,r) - 2d(a,r)$.
It's easy to see that all the $d(i,r)$ can be computed in $O(n)$ steps with BFS. There's also a data structure of Harel and Tarjan that, after $O(n)$ preprocessing will answer least common ancestor queries in $O(1)$ time. So the whole thing becomes $O(n^2)$.
REPLY [1 votes]: Another variant would be to start with an arbitrary vertex and then to update the all pairs shortest paths table each time a new vertex is discovered, that is adjacent to one of the previously discovered ones.
When discovering vertex $v$ via edge $(u,v)$, the entries of row $u$ and of column $v$ of the distance table $T$ are first duplicated into row $v$ and colum $v$ respectively; then to every newly generated non-zero entry in the distance table the weight $w(u,v)$ of edge $(u,v)$ is added and finally the distance from $u$ to $v$ and from $v$ to $u$ are set to $w(u,v)$
The sequential complexity is $O(n^2)$, but the algorithm is much simpler and in contrast to the other variants, this algorithm also works for dynamically growing trees, i.e. initially unknown trees and/or trees with no upper bound on final size.
That algorithm can also exploit parallel computing capability, e.g. of GPUs; the complexity would then be $O(n)$ under the assumption that an unlimited number of processors is available. | 88,404 |
Most.
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TITLE: Complex normal coordinates in Kähler manifolds
QUESTION [6 upvotes]: Let $(M, g, J, \omega)$ be a Kähler manifold. That is, $(M, J)$ is a complex manifold, $g$ is a Hermitian metric on $M$ and
$$\omega (X, Y) = g(JX, Y)$$
is a closed two form.
As a Riemannian manifold $(M, g)$, for each $x\in M$, one can find a geodesic normal coordinates around each $x$. In the case of Kähler metric, one actually have more:
Proposition: (Complex Normal Coordinates on Kähler manifolds) For each $x\in M$, there is a local holomorphic coordinates around $x$ so that the metric $g = g_{i\bar j}$ satisfies
$$g_{i\bar j}(x) = \delta_{ij}, \ \ d g_{i\bar j} (x) = 0, \ \ \ \frac{\partial^2 g_{i\bar j}}{\partial z_k \partial z_l} (x) = 0.$$
While the first two conditions are similar to what we have for geodesic normal coordinates in Riemannian geometry, there is no corresponding analogue for the last condition. Also, even in a Kähler manifold, the geodesic normal coordinates might not be holomorphic.
I am looking for a proof of this proposition.
REPLY [4 votes]: The result is local, we may assume that $p =0 \in U$, where $U \subset \mathbb C^n$ is an open set. By an complex linear transformation, we may also assume $h_{i\bar j}(0) = \delta_{ij}$. Consider the holomorphic mapping $\phi : B_\epsilon \to U$, given by
$$z_i=\phi (w)_i = w_i + A^i_{mn} w_mw_n + B^i_{pqr} w_pw_qw_r,$$
here $A^i_{mn}, B^i_{pqr}$ are constants to be chosen, and repeated indices means sum over. We also assume that $A^i_{mn}, B^i_{pqr}$ are fully symmetric with respect to the lower indices. Since $d\phi_0 = Id$, $\phi$ is an biholomorphism onto its image (restricting to smaller sets when necessary). Let $g = \phi^* h$ be the pullback metric. Then using the new cordinates $(w_1, \cdots, w_n)$,
\begin{align}\tag{1} g_{\alpha \bar \beta} &= \frac{\partial z_i}{\partial w^\alpha}\overline{\frac{\partial z_j}{\partial w^\beta}}h_{i\bar j}
\end{align}
and
$$\tag{2} \frac{\partial z_i}{\partial w_\alpha} = \delta_{i\alpha} + 2A^i_{\alpha m} w_m + 3B^i_{\alpha pq } w_pw_q$$
Then we have
\begin{align}g_{\alpha\bar\beta} &= h_{\alpha\bar\beta} +2A^i_{\alpha m} h_{i\bar\beta}w_n + 2\overline{A^j_{\beta n}} h_{\alpha \bar j} \bar w_n \\
&\ \ \ + 3B^i_{\alpha pq } w_pw_q h_{i\bar\beta} + 3\overline{B^j_{\beta rs} } \bar w_r \bar w_s h_{\alpha \bar j} + 4 A^i_{\alpha m} \overline{A^j_{\beta n} } h_{i\bar j}w_m \bar w_n\\
&\ \ \ + O(|w|^3).
\end{align}
From here it's obvious that $g_{\alpha\bar\beta}(0) = \delta_{\alpha\beta}$. Also, we have
\begin{align} \partial_\gamma g_{\alpha\bar\beta} (0) &= \partial_\gamma h_{\alpha\bar\beta} (0) + 2A^\beta_{\alpha\gamma}, \\
\bar\partial_\gamma g_{\alpha\bar\beta} (0) &= \bar\partial_\gamma h_{\alpha\bar\beta} (0) + 2\overline{A^\alpha_{\beta\gamma}}, \\
\partial_\eta \partial _\gamma g_{\alpha\bar\beta} (0) &= \partial_\eta \partial _\gamma h_{\alpha\bar\beta} (0) + 6B^\beta_{\alpha\gamma\eta}.
\end{align}
Now we choose $A^\beta_{\alpha\gamma} = -\frac 12 \partial_\gamma h_{\alpha\bar\beta}$. First of all, $A$ so chosen is really symmetric in the lower indice, since
$$\tag{3}\partial_\gamma h_{\alpha\bar\beta} =\partial_\alpha h_{\gamma \bar\beta}$$
when $h$ is a Kähler metric (see here). Next, since $h$ is Hermitian,
$$\bar \partial_\gamma h_{\alpha\bar\beta} (0) = \overline{\partial_\gamma \overline{h_{\alpha\bar\beta}(0)}} = \overline{\partial_\gamma h_{\beta\bar\alpha}(0)} = -\frac{1}{2} \overline{A^\alpha_{\beta\gamma}},
$$
where in the last equality we used the definition of $A$. Thus we have $dg_{i\bar j}(0) = 0$.
Lastly, we choose $B^\beta_{\alpha\gamma\delta} = - \partial_\eta \partial _\gamma h_{\alpha\bar\eta} (0)$. Again by (3), $\partial_\eta \partial _\gamma h_{\alpha\bar\eta}$ is symmetric in $\alpha, \gamma, \eta$. Thus $B$ is again well defined, and this finishes the proof.
In this book, they state the following proposition:
Proposition 1.6: (Normal coordinates in Kähler case) Let $M$ be a Kähler manifold with a real analytic Kähler metric. Given $x\in M$, there exist local complex coordinates $(z_1, \cdots, z_n)$ unique modulo unitary linear transformations such that $g_{i\bar j}(x) = \delta_{ij}$, $dg_{i\bar j}(x)= 0$ and
$$ \frac{\partial ^l g_{i\bar j}}{\partial z_{i_1} \cdots \partial z_{i_k}} (x) = 0$$
for all $l\ge 0$ and $i_1 + \cdots + i_k = l$, and this also holds for its conjugate.
They also suggest a reference, p.286, claiming that there's an elegant proof. | 64,018 |
\begin{document}
\begin{center}
{\large\bf A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$}
\end{center}
\vskip 2mm \centerline{Ji-Cai Liu}
\begin{center}
{\footnotesize Department of Mathematics, Wenzhou University, Wenzhou 325035, PR China\\
{\tt [email protected] } \\[10pt]
}
\end{center}
\vskip 0.7cm \noindent{\bf Abstract.}
We prove three supercongruences for sums of Almkvist--Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruence:
\begin{align*}
\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\gamma_k \equiv \left(\frac{-3}{p}\right)p\pmod{p^3},
\end{align*}
which is corresponding to Chan and Verrill's formula for $1/\pi$:
\begin{align*}
\sum_{k=0}^{\infty}\frac{4k+1}{81^k}\gamma_k=\frac{3\sqrt{3}}{2\pi}.
\end{align*}
Here $\gamma_n$ are the Almkvist--Zudilin numbers.
\vskip 3mm \noindent {\it Keywords}: Supercongruences; Almkvist--Zudilin numbers; Harmonic numbers
\vskip 2mm
\noindent{\it MR Subject Classifications}: 11A07, 11B65, 11Y55, 05A19
\section{Introduction}
For $n\ge 0$, the following sequence:
\begin{align*}
\gamma_n=\sum_{j=0}^{n}(-1)^{n-j}\frac{3^{n-3j}(3j)!}{(j!)^3}{n\choose 3j}{n+j\choose j}
\end{align*}
are known as Almkvist--Zudilin numbers (see \cite{az-b-2007} and A125143 in \cite{sloane-2020}).
This sequence appears to be first recorded by Zagier \cite{zagier-b-2009} as integral solutions to Ap\'ery-like recurrence equations.
These numbers also appear as coefficients of modular forms.
Let $q=e^{2\pi i \tau}$ and
\begin{align*}
\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)
\end{align*}
be the Dedekind eta function. Chan and Verrill \cite{cv-mrl-2009} showed that if
\begin{align*}
t_3(\tau)=\left(\frac{\eta(3\tau)\eta(6\tau)}{\eta(\tau)\eta(2\tau)}\right)^4\quad
\text{and}\quad F_3(\tau)=\frac{\left(\eta(\tau)\eta(2\tau)\right)^3}{\eta(3\tau)\eta(6\tau)},
\end{align*}
and $|t_3(\tau)|$ is sufficiently small, then
\begin{align*}
F_3(\tau)=\sum_{n=0}^{\infty}\gamma_n t_3^n(\tau).
\end{align*}
They also constructed some new series for $1/\pi$ in terms of the numbers $\gamma_n$, one of the typical examples is the following formula \cite[Theorem 3.14]{cv-mrl-2009}:
\begin{align}
\sum_{k=0}^{\infty}\frac{4k+1}{81^k}\gamma_k=\frac{3\sqrt{3}}{2\pi}.\label{aa-1}
\end{align}
The above interesting example motivates us to prove the following supercongruence, which was originally conjectured by Zudilin \cite[(33)]{zudilin-jnt-2009}.
\begin{thm}\label{t-1}
For any prime $p\ge 5$, we have
\begin{align}
\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\gamma_k \equiv \left(\frac{-3}{p}\right)p\pmod{p^3},\label{aa-2}
\end{align}
where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol.
\end{thm}
The supercongruence \eqref{aa-2} may be regarded as a $p$-adic analogue of \eqref{aa-1}. In the past two decades, Ramanujan-type series for $1/\pi$ as well as related supercongruences and $q$-supercongruences have attracted many experts' attention (see, for instance, \cite{ccl-am-2004,cv-mrl-2009,cwz-ijm-2013,guo-aam-2020,gl-itsf-2020,gs-rm-2020,
gz-am-2019,liu-jsc-2019,liu-jmaa-2020,lp-a-2020,sunzw-scm-2011,sunzw-era-2020,van-b-1997,zudilin-jnt-2009}).
The second result of this paper consists of the following two related supercongruences involving
the numbers $\gamma_n$, which were originally conjectured by Z.-H. Sun \cite[Conjecture 6.8]{sunzh-itsf-2015}.
\begin{thm}\label{t-2}
For any prime $p\ge 5$, we have
\begin{align}
\sum_{k=0}^{p-1}(4k+3)\gamma_k \equiv 3\left(\frac{-3}{p}\right)p\pmod{p^3}.\label{aa-3}
\end{align}
\end{thm}
\begin{thm}\label{t-3}
For any prime $p\ge 5$, we have
\begin{align}
\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\gamma_k\equiv \left(\frac{-3}{p}\right)p\pmod{p^3}.\label{aa-4}
\end{align}
\end{thm}
We remark that congruence properties for the Almkvist--Zudilin numbers have been widely investigated by Amdeberhan and Tauraso \cite{at-aa-2016}, Chan, Cooper and Sica \cite{ccs-ijnt-2010}, and Z.-H. Sun \cite{sunzh-itsf-2015,sunzh-a-20-2,sunzh-a-20-4}.
The rest of the paper is organized as follows. In Section 2, we recall some necessary combinatorial identities involving harmonic numbers and prove a preliminary congruence. The proofs of Theorems \ref{t-1}--\ref{t-3} are presented in Sections 3--5, respectively.
\section{Preliminary results}
Let
\begin{align*}
H_n=\sum_{j=1}^n\frac{1}{j}
\end{align*}
denote the $n$th harmonic number. The Fermat quotient of an integer $a$ with respect to an odd prime $p$ is given by $q_p(a)=(a^{p-1}-1)/p$.
In order to prove Theorems \ref{t-1} and \ref{t-2}, we need the following two lemmas.
\begin{lem}
For any non-negative integer $n$, we have
\begin{align}
&\sum_{i=0}^{n}(-1)^i{n\choose i}{n+i\choose i}=(-1)^n,\label{bb-1}\\
&\sum_{i=0}^{n}(-1)^i{n\choose i}{n+i\choose i}H_i=2(-1)^nH_n,\label{bb-2}\\
&\sum_{i=0}^{n}(-1)^i{n\choose i}{n+i\choose i}H_{n+i}=2(-1)^nH_n.\label{bb-3}
\end{align}
\end{lem}
In fact, such identities can be discovered and proved by the symbolic summation package {\tt Sigma} developed by Schneider \cite{schneider-slc-2007}. One can also refer to \cite{liu-jsc-2019} for the same approach to finding and proving identities of this type. For human proofs of \eqref{bb-1}--\eqref{bb-3}, one refers to \cite{prodinger-integers-2008}.
\begin{lem}
For any prime $p\ge 5$, we have
\begin{align}
\sum_{k=0}^{p-1}
\frac{(3k)!}{3^{3k} k!^3}\left(H_{3k}-H_k\right)\equiv \left(\frac{-3}{p}\right)q_p(3)\pmod{p}.
\label{bb-4}
\end{align}
\end{lem}
{\noindent\it Proof.}
Note that
\begin{align}
\sum_{k=0}^{p-1}
\frac{(3k)!}{3^{3k} k!^3}\left(3H_{3k}-H_k\right)
=\sum_{k=0}^{p-1}\frac{(1/3)_k(2/3)_k}{(1)_k^2}\sum_{j=0}^{k-1}
\left(\frac{1}{1/3+j}+\frac{1}{2/3+j}\right).
\label{bb-5}
\end{align}
Recall the following identity due to Tauraso \cite[Theorem 1]{tauraso-integers-2012}:
\begin{align}
\frac{(1/3)_k(2/3)_k}{(1)_k^2}\sum_{j=0}^{k-1}\left(\frac{1}{1/3+j}+\frac{1}{2/3+j}\right)
=\sum_{j=0}^{k-1}\frac{(1/3)_j(2/3)_j}{(1)_j^2}\cdot \frac{1}{k-j}.
\label{bb-6}
\end{align}
Substituting \eqref{bb-6} into \eqref{bb-5} and exchanging the summation order gives
\begin{align}
\sum_{k=0}^{p-1}
\frac{(3k)!}{3^{3k} k!^3}\left(3H_{3k}-H_k\right)
&=\sum_{k=0}^{p-1}\sum_{j=0}^{k-1}\frac{(1/3)_j(2/3)_j}{(1)_j^2}\cdot \frac{1}{k-j}\notag\\[10pt]
&=\sum_{j=0}^{p-2}\frac{(1/3)_j(2/3)_j}{(1)_j^2}H_{p-1-j}\notag\\[10pt]
&\equiv \sum_{j=0}^{p-2}\frac{(3j)!}{3^{3j} j!^3}H_{j}\pmod{p},\label{bb-7}
\end{align}
where we have utilized the fact that $H_{p-1-j}\equiv H_j\pmod{p}$.
By \eqref{bb-7}, we obtain
\begin{align}
\sum_{k=0}^{p-1}\frac{(3k)!}{3^{3k} k!^3}\left(H_{3k}-H_k\right)
&\equiv \frac{1}{3}\left(\sum_{k=0}^{p-2}\frac{(3k)!}{3^{3k} k!^3}H_{k}-2\sum_{k=0}^{p-1}\frac{(3k)!}{3^{3k} k!^3}H_k\right)\notag\\[10pt]
&\equiv -\frac{1}{3}\sum_{k=0}^{\lfloor p/3\rfloor}\frac{(3k)!}{3^{3k} k!^3}H_{k} \pmod{p},
\label{bb-8}
\end{align}
because $(3k)!\equiv 0\pmod{p}$ for $k>\lfloor p/3\rfloor$.
Let $m=\lfloor p/3\rfloor$. From \cite[Lemma 2.3]{at-aa-2016}, we see that for $0\le k\le m$,
\begin{align}
\frac{(3k)!}{3^{3k} k!^3}\equiv (-1)^k{m\choose k}{m+k\choose k}\pmod{p}.
\label{bb-9}
\end{align}
It follows from \eqref{bb-2}, \eqref{bb-8} and \eqref{bb-9} that
\begin{align*}
\sum_{k=0}^{p-1}\frac{(3k)!}{3^{3k} k!^3}\left(H_{3k}-H_k\right)
&\equiv -\frac{1}{3}\sum_{k=0}^m(-1)^k{m\choose k}{m+k\choose k}H_k\pmod{p}\\[10pt]
&=-\frac{2(-1)^m}{3}H_m.
\end{align*}
Finally, noting
\begin{align}
(-1)^{\lfloor p/3\rfloor}=\left(\frac{-3}{p}\right),\label{bb-10}
\end{align}
and the following congruence \cite[page 359]{lehmer-am-1938}:
\begin{align}
H_{\lfloor p/3\rfloor}\equiv -\frac{3}{2}q_p(3) \pmod{p^2},\label{bb-11}
\end{align}
we complete the proof of \eqref{bb-4}.
\qed
\section{Proof of Theorem \ref{t-1}}
We begin with the transformation formula due to Chan and Zudilin \cite[Corollary 4.3]{cz-m-2010}:
\begin{align}
\gamma_n=\sum_{i=0}^n{2i\choose i}^2{4i\choose 2i}{n+3i\choose 4i}(-3)^{3(n-i)}.\label{cc-1}
\end{align}
Using \eqref{cc-1} and exchanging the summation order, we obtain
\begin{align}
\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\gamma_k
&=\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\sum_{i=0}^k{2i\choose i}^2{4i\choose 2i}{k+3i\choose 4i}(-3)^{3(k-i)}\notag\\[10pt]
&=\sum_{i=0}^{p-1}\frac{1}{(-3)^{3i}}{2i\choose i}^2{4i\choose 2i}\sum_{k=i}^{p-1}\frac{4k+1}{(-3)^k}{k+3i\choose 4i}.\label{cc-2}
\end{align}
Note that
\begin{align}
\sum_{k=i}^{n-1}\frac{4k+1}{(-3)^k}{k+3i\choose 4i}=(n-i){n+3i\choose 4i}(-3)^{1-n},\label{cc-3}
\end{align}
which can be easily proved by induction on $n$. Combining \eqref{cc-2} and \eqref{cc-3} gives
\begin{align}
\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\gamma_k
=3^{1-p}\sum_{i=0}^{p-1}\frac{p-i}{(-3)^{3i}}{2i\choose i}^2{4i\choose 2i}{p+3i\choose 4i}.\label{cc-4}
\end{align}
Furthermore, we have
\begin{align*}
&(-1)^i(p-i){2i\choose i}^2{4i\choose 2i}{p+3i\choose 4i}\\[10pt]
&=\frac{(-1)^ip(p+3i)\cdots (p+1)(p-1)\cdots (p-i)}{i!^4}\\[10pt]
&\equiv \frac{p(3i)!}{i!^3}\left(1+p\left(H_{3i}-H_i\right)\right)\pmod{p^3}.
\end{align*}
Thus,
\begin{align*}
\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\gamma_k
&\equiv 3^{1-p}p\sum_{i=0}^{p-1}
\frac{(3i)!}{3^{3i} i!^3}\left(1+p\left(H_{3i}-H_i\right)\right)\pmod{p^3}.
\end{align*}
Finally, noting \eqref{bb-4} and Mortenson's supercongruence \cite[(1.2)]{mortenson-tams-2003}:
\begin{align*}
\sum_{i=0}^{p-1}
\frac{(3i)!}{3^{3i} i!^3}
\equiv \left(\frac{-3}{p}\right) \pmod{p^2},
\end{align*}
we arrive at
\begin{align*}
\sum_{k=0}^{p-1}\frac{4k+1}{81^k}\gamma_k
&\equiv p\left(\frac{-3}{p}\right)\left(3^{1-p}+3^{1-p}pq_p(3)\right)\pmod{p^3}\\[10pt]
&=p\left(\frac{-3}{p}\right),
\end{align*}
as desired.
\section{Proof of Theorem \ref{t-2}}
Recall the following transformation formula \cite[(5.1)]{sunzh-itsf-2015}:
\begin{align}
\gamma_n=\sum_{i=0}^{\lfloor n/3 \rfloor}{2i\choose i}^2{4i\choose 2i}{n+i\choose 4i}(-3)^{n-3i}.\label{dd-1}
\end{align}
By \eqref{dd-1}, we have
\begin{align}
\sum_{k=0}^{p-1}(4k+3)\gamma_k
&=\sum_{k=0}^{p-1}(4k+3)\sum_{i=0}^{\lfloor k/3 \rfloor}{2i\choose i}^2{4i\choose 2i}{k+i\choose 4i}(-3)^{k-3i}\notag\\[10pt]
&=\sum_{i=0}^{p-1}\frac{1}{(-3)^{3i}}{2i\choose i}^2{4i\choose 2i}\sum_{k=i}^{p-1}(-3)^k(4k+3){k+i\choose 4i}.\label{dd-2}
\end{align}
It can be easily proved by induction on $n$ that
\begin{align}
\sum_{k=i}^{n-1}(-3)^k(4k+3){k+i\choose 4i}=3(n-3i){n+i\choose 4i}(-3)^{n-1}.\label{dd-3}
\end{align}
It follows from \eqref{dd-2} and \eqref{dd-3} that
\begin{align*}
\sum_{k=0}^{p-1}(4k+3)\gamma_k
=3^{p}\sum_{i=0}^{\lfloor p/3\rfloor}\frac{p-3i}{(-3)^{3i}}{2i\choose i}^2{4i\choose 2i}{p+i\choose 4i}.
\end{align*}
Note that
\begin{align*}
&(-1)^i(p-3i){2i\choose i}^2{4i\choose 2i}{p+i\choose 4i}\\[10pt]
&=\frac{(-1)^ip(p+i)\cdots(p+1)(p-1)\cdots(p-3i)}{i!^4}\\[10pt]
&\equiv \frac{p(3i)!}{i!^3}\left(1-p\left(H_{3i}-H_i\right)\right)\pmod{p^3}.
\end{align*}
Thus,
\begin{align}
\sum_{k=0}^{p-1}(4k+3)\gamma_k
&\equiv 3^{p}p
\sum_{i=0}^{\lfloor p/3\rfloor}\frac{(3i)!}{3^{3i} i!^3}\left(1-p\left(H_{3i}-H_i\right)\right)\pmod{p^3}.\label{dd-4}
\end{align}
Let $m=\lfloor p/3\rfloor$. Since
\begin{align*}
\frac{(3i)!}{3^{3i} i!^3}=(-1)^i{-1/3\choose i}{-1/3+i\choose i}
=(-1)^i{-2/3\choose i}{-2/3+i\choose i},
\end{align*}
we have
\begin{align}
\frac{(3i)!}{3^{3i} i!^3}
&=(-1)^i{m-p/3\choose i}{m-p/3+i\choose i}\notag\\[10pt]
&=\frac{(-1)^i(m+i-p/3)\cdots (m-i+1-p/3)}{i!^2}\notag\\[10pt]
&\equiv (-1)^i{m\choose i}{m+i\choose i}\left(1-\frac{p}{3}\left(H_{m+i}-H_{m-i}\right)\right)\pmod{p^2}.
\label{dd-5}
\end{align}
Substituting \eqref{dd-5} into the right-hand side of \eqref{dd-4} gives
\begin{align}
&\sum_{k=0}^{p-1}(4k+3)\gamma_k\notag\\
&\equiv 3^{p}p
\sum_{i=0}^{m}(-1)^i{m\choose i}{m+i\choose i}\notag\\
&\times \left(1-\frac{p}{3}\left(H_{m+i}-H_{m-i}+3H_{3i}-3H_i\right)\right)
\pmod{p^3}.\label{dd-6}
\end{align}
Furthermore, we have
\begin{align}
H_{3i}&=\frac{1}{3}\left(H_i+\sum_{j=1}^i\frac{1}{j-1/3}+\sum_{j=1}^i\frac{1}{j-2/3}\right)\notag\\[10pt]
&\equiv \frac{1}{3}\left(H_i+\sum_{j=1}^i\frac{1}{m+j}-\sum_{j=1}^i\frac{1}{m+1-j}\right)\pmod{p}\notag\\[10pt]
&=\frac{1}{3}\left(H_i+H_{m+i}+H_{m-i}-2H_m\right).\label{dd-7}
\end{align}
It follows from \eqref{bb-1}--\eqref{bb-3}, \eqref{dd-6} and \eqref{dd-7} that
\begin{align*}
&\sum_{k=0}^{p-1}(4k+3)\gamma_k\\
&\equiv 3^{p}p
\sum_{i=0}^{m}(-1)^i{m\choose i}{m+i\choose i}\left(1-\frac{2p}{3}\left(H_{m+i}-H_m-H_i\right)\right)
\pmod{p^3}\\
&=3^{p}p(-1)^m\left(1+\frac{2p}{3}H_m\right).
\end{align*}
Finally, using \eqref{bb-10} and \eqref{bb-11}, we obtain
\begin{align*}
\sum_{k=0}^{p-1}(4k+3)\gamma_k
&\equiv 3p\left(\frac{-3}{p}\right)3^{p-1}\left(2-3^{p-1}\right)\\[10pt]
&=3p\left(\frac{-3}{p}\right)\left(1-\left(3^{p-1}-1\right)^2\right)\\[10pt]
&\equiv 3p\left(\frac{-3}{p}\right)\pmod{p^3},
\end{align*}
where we have used the Fermat's little theorem in the last step.
\section{Proof of Theorem \ref{t-3}}
Recall the following transformation formula \cite[Lemma 4.1]{sunzh-itsf-2015}:
\begin{align}
\gamma_n=\sum_{i=0}^{n}(-9)^{n-i}{2i\choose i}{n+i\choose 2i}\sum_{j=0}^i{i\choose j}^2{2j\choose j}.\label{ee-1}
\end{align}
Let
\begin{align*}
g_n=\sum_{k=0}^n{n\choose k}^2{2k\choose k}.
\end{align*}
By \eqref{ee-1}, we have
\begin{align}
\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\gamma_k
&=\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\sum_{i=0}^{k}(-9)^{k-i}{2i\choose i}{k+i\choose 2i}g_i\notag\\[10pt]
&=\sum_{i=0}^{p-1}\frac{g_i}{(-9)^i}\sum_{k=i}^{p-1}(2k+1){k+i\choose 2i}{2i\choose i}.\label{ee-2}
\end{align}
Note that
\begin{align}
\sum_{k=i}^{n-1}(2k+1){k+i\choose 2i}{2i \choose i}
=\frac{n^2}{i+1}{n-1\choose i}{n+i\choose i},\label{ee-3}
\end{align}
which can be proved by induction on $n$. It follows from \eqref{ee-2} and \eqref{ee-3} that
\begin{align*}
\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\gamma_k
=p^2\sum_{i=0}^{p-1}\frac{g_i}{(-9)^i(i+1)} {p-1\choose i}{p+i\choose i}.
\end{align*}
Since
\begin{align*}
{p-1\choose i}{p+i\choose i}\equiv (-1)^i\pmod{p^2},
\end{align*}
we have
\begin{align}
\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\gamma_k\equiv p^2\sum_{i=0}^{p-1}\frac{g_i}{9^i(i+1)}\pmod{p^3}.
\label{ee-4}
\end{align}
From \cite[Lemma 2.7]{jv-rj-2010}, we see that for $0\le i\le p-1$,
\begin{align*}
\frac{g_i}{9^i}\equiv \left(\frac{-3}{p}\right)g_{p-1-i}\pmod{p},
\end{align*}
and so
\begin{align*}
\sum_{i=0}^{p-2}\frac{g_i}{9^i(i+1)}
&\equiv \left(\frac{-3}{p}\right)\sum_{i=0}^{p-2}\frac{g_{p-1-i}}{i+1}\\[10pt]
&=\left(\frac{-3}{p}\right)\sum_{i=1}^{p-1}\frac{g_{i}}{p-i}\\[10pt]
&\equiv -\left(\frac{-3}{p}\right)\sum_{i=1}^{p-1}\frac{g_{i}}{i}\pmod{p}.
\end{align*}
Using the congruence \cite[(1.8)]{sunzw-rj-2016}:
\begin{align*}
\sum_{i=1}^{p-1}\frac{g_{i}}{i}\equiv 0\pmod{p},
\end{align*}
we obtain
\begin{align}
\sum_{i=0}^{p-2}\frac{g_i}{9^i(i+1)}\equiv 0\pmod{p}.\label{ee-5}
\end{align}
Furthermore, combining \eqref{ee-4} and \eqref{ee-5} gives
\begin{align*}
\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\gamma_k\equiv \frac{pg_{p-1}}{9^{p-1}}\pmod{p^3}.
\end{align*}
By \cite[Lemma 3.2]{sunzw-rj-2016}, we have
\begin{align*}
g_{p-1}\equiv \left(\frac{-3}{p}\right)\left(2\cdot3^{p-1}-1\right)\pmod{p^2},
\end{align*}
and so
\begin{align*}
\sum_{k=0}^{p-1}\frac{2k+1}{(-9)^k}\gamma_k
&\equiv p\left(\frac{-3}{p}\right)
\left(1-\frac{\left(3^{p-1}-1\right)^2}{9^{p-1}}\right)\\[10pt]
&\equiv p\left(\frac{-3}{p}\right)\pmod{p^3},
\end{align*}
where we have utilized the Fermat's little theorem.
{\noindent\bf Remark.}
Z.-H. Sun \cite[Conjecture 6.8]{sunzh-itsf-2015} also conjectured a companion supercongruence of
\eqref{aa-4}:
\begin{align}
\sum_{k=0}^{p-1}\frac{2k+1}{9^k}\gamma_k \equiv \left(\frac{-3}{p}\right)p\pmod{p^3}.
\label{ee-6}
\end{align}
In a similar way, by using \eqref{ee-1} and the following identity:
\begin{align*}
\sum_{k=i}^{n-1}(-1)^k(2k+1){k+i\choose 2i}{2i \choose i}
=(-1)^{n-1}n{n-1\choose i}{n+i\choose i},
\end{align*}
we can show that
\begin{align*}
\sum_{k=0}^{p-1}\frac{2k+1}{9^k}\gamma_k
\equiv p\sum_{i=0}^{p-1}\frac{g_i}{9^i}\pmod{p^3}.
\end{align*}
Thus, the conjectural supercongruence \eqref{ee-6} is equivalent to
\begin{align*}
\sum_{i=0}^{p-1}\frac{g_i}{9^i}\equiv \left(\frac{-3}{p}\right)\pmod{p^2},
\end{align*}
which was originally conjectured by Z.-W. Sun \cite[Remark 1.1]{sunzw-rj-2016}.
\vskip 5mm \noindent{\bf Acknowledgments.}
This work was supported by the National Natural Science Foundation of China (grant 11801417). | 104,268 |
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Local in the June rioting in Maungdaw and Buthidaung townships.
A group of Rakhine victims after their homes are burned down by the Bangladeshi immigrants
According to a police officer from Buthidaung, all legal proceedings including investigation have already done and the accused are only waiting for the decision when they receive the release order. It is said that Saw Myint (a) Mamud, Zaw Myint and Than Naing (a) Adu High of UNHCR from Buthidaung and Win Naing and Kyaw Hla Aung of Doctors Without Borders (MSF) from Sittwe prison are among the 10 terrorist suspects who were suddenly released by the Government without any official explanation. "The release order is signed by an officer named Pyone Cho from the Home Ministry affairs" the officer added. A Buthidaung resident who is also a lawyer told RMG that he didn't believe the suspects can be released easily like this without a special diplomatic pressure from one of the world's super power country like United States. Since they belong to illegally migrated Bangladeshi Muslim community and not the US citizens, the UN and INGO operatives must have paid a large amount of money to the officials. I am sure that they (UN and INGO officials) exploited our country's unstable law and order situation, and bad governance in helping terrorists."
Illegal Bangladeshi immigrants and terrorist suspects Kyaw Hla Aung and Win Naing in front of Sthe ittwe prison gate. (photo-Mizzima)
It is reported that the accused Bengali Muslim migrants used their organizational computers and internet facilities to communicate with the Muslim extremist groups from Pakistan, UK and Middle Eastern countries. Their irresponsible propagation of false Muslim massacre by the local Buddhist Rakhine and Myanmar armed security forces led an uncontrollable community riot between local Buddhist Rakhine and Bengali Muslim immigrants. The said violence, according to Myanmar government report, have resulted 31 Rakhines and 46 Bengalis dead and more than four thousand houses burned down.
The UNHCR and INGO organizations that are working in Rakhine state have a long history of discriminating against the local people as they are only giving a special benefits to the Bengali immigrants while totally neglecting local Rakhine and other Myanmar ethnic people. As a result of this, the Rakhine and other ethnic victims from all over Rakhine state denied the assistance offered to them by the UN and INGOs after many Rakhine died and thousands of their houses are burned down by the Bengali migrants in the riot. "The government has a weakness in implementing law and order of the country. We are glad to hear that a man is freed from the prison.", the Rakhine Nationalities Development Party (RNDP) general secretary U U Hla Saw told to Narinjara News in an interview. Those accused are somehow involved in the riot. No one should be above the law but the release of the accused without the punishment is a sad news for our Rakhine people", he continued.
UN and INGOs are being accused of racial discrimination and demanded by local ethnic people to get out of their land
Many Myanmar citizens including Myanmar people living around the world are also concerned about government's decision to release the terrorist suspects, as this move will encourage Bangladeshi illegal immigrants who want to create a new Islamic country, to create more problems against the local ethnic people. | 307,585 |
In this section, we take a closer look at the geometry of $\mathfrak X$, and compare it with related moduli spaces.
\subsection{Comparison of $\mathfrak{X}$ with the spaces of weighted admissible covers $\overline{\mathcal{H}}_4^3(1/6+\epsilon)$}
Recall that $\overline{\mathcal H}_4^3(1/6 + \epsilon)$ is the moduli space of weighted admissible covers where up to 5 branch points are allowed to coincide.
Let $U \subset \overline{\mathcal H}_4^3(1/6 + \epsilon)$ be the open substack parametrizing $\phi \from C \to P$ where $P \cong \PP^1$, the curve $C$ is smooth, and the Tschirnhausen bundle $E_\phi$ of $\phi$ is $\O(3) \oplus \O(3)$.
We have a morphism $\Phi \from U \to \mathfrak X$ given by the transformation
\[ (\phi \from C \to P) \mapsto (\PP E_\phi, C).\]
\begin{theorem}\label{thm:phi_extends}
The map $\Phi$ extends to a morphism of stacks $\Phi \from \overline{\mathcal{H}}_4^3(1/6+\epsilon) \to \mathfrak X$.
\end{theorem}
Since $\overline{\mathcal{H}}_4^3(1/6+\epsilon)$ is proper and $\mathfrak X$ is separated, the map $\Phi$ is also proper.
For the proof, we need extension lemmas for morphisms of stacks, extending some well-known results for schemes.
Let $\mathcal X$ and $\mathcal Y$ be separated Deligne--Mumford stacks of finite type over a field $\k$.
Let $\mathcal U \subset \mathcal X$ be a dense open substack.
\begin{lemma}\label{lem:unique_extension}
Assume that $\mathcal X$ is normal.
If $\Phi_1, \Phi_2 \from \mathcal X \to \mathcal Y$ are two morphisms whose restrictions to $\mathcal U$ are equal (2-isomorphic), then $\Phi_1$ and $\Phi_2$ are equal (2-isomorphic).
\end{lemma}
\begin{proof}
We have the following diagram where the square is a pull-back
\[
\begin{tikzpicture}
\node (Y) {$\mathcal Y$};
\node[below of=Y] (YY) {$\mathcal Y \times \mathcal Y$};
\draw (YY) ++ (-2cm,0) node (X) {$\mathcal X$};
\node (U) [left of=X] {$\mathcal U$};
\node[above of=X] (Z) {$\mathcal Z$};
\draw[->] (Z) edge (Y) (Z) edge (X) (U) edge (X) (X) edge node[above] {\tiny $(\Phi_1, \Phi_2)$} (YY) (Y) edge node[right] {\tiny $\Delta$} (YY);
\end{tikzpicture}
\]
Since $\mathcal Y$ is a separated Deligne--Mumford stack, the diagonal map $\Delta$ is representable, proper, and unramified.
Therefore, so is the pullback $\mathcal Z \to \mathcal X$.
Since $\Phi_1$ and $\Phi_2$ agree on $\mathcal U$, the inclusion $\mathcal U \to \mathcal X$ lifts to $\mathcal U \to \mathcal Z$.
Since $\mathcal Z \to \mathcal X$ is unramified and $\mathcal U \to \mathcal X$ is an open immersion, so is the lift $\mathcal U \to \mathcal Z$.
Let $\overline {\mathcal U} \subset \mathcal Z$ be the closure of $\mathcal U$ and $\overline{\mathcal U}^\nu \to \overline{\mathcal U}$ its normalization.
Since $\mathcal X$ is normal, Zariski's main theorem implies that $\overline {\mathcal U}^\nu \to \mathcal X$ is an isomorphism.
Hence the map $\mathcal Z \to \mathcal X$ admits a section $\mathcal X \to \mathcal Z$.
In other words, the map $(\Phi_1, \Phi_2) \from \mathcal X \to \mathcal Y \times \mathcal Y$ factors through the diagonal $\mathcal Y \to \mathcal Y \times \mathcal Y$.
\end{proof}
\begin{example}
In \autoref{lem:unique_extension}, we can drop the normality assumption on $\mathcal X$ if $\mathcal Y$ is an algebraic space, but not otherwise.
An example of distinct maps that agree on a dense open substack can be constructed using twisted curves (see \cite[Proposition~7.1.1]{abr.cor.vis:03}).
Let $\mathcal X$ be the stack
\[ \mathcal X = \left[ \spec \C[x,y]/xy / \mu_n \right],\]
where $\zeta \in \mu_n$ acts by $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$.
Every $\zeta \in \mu_n$ defines an automorphism of $t_\zeta \from \mathcal X \to \mathcal X$ given by $(x,y) \mapsto (x, \zeta y)$.
The map $t_\zeta$ is the identity map on the complement of the node of $\mathcal X$, but not the identity map on $\mathcal X$ if $\zeta \neq 1$.
\end{example}
The map $\Phi \from \mathcal U \to \mathcal Y$ induces a map $|\Phi| \from |\mathcal U| \to |\mathcal Y|$ on the set of points.
Let $\phi \from |\mathcal X| \to |\mathcal Y|$ be an extension of $|\Phi|$.
We say that $\phi$ is \emph{continuous in one-parameter families} if for every DVR $\Delta$ and a map $i \from \Delta \to \mathcal X$ that sends the generic point $\eta$ of $\Delta$ to $\mathcal U$, the map $\Phi \circ i \from \eta \to \mathcal Y$ extends to a map $\Delta \to \mathcal Y$ and agrees with the map $\phi$ on the special point.
\begin{lemma}\label{lem:exists_extension}
Suppose $\mathcal X$ is smooth, and $\Phi \from \mathcal U \to \mathcal Y$ is a morphism.
Let $\phi \from |\mathcal X| \to |\mathcal Y|$ be an extension of $|\Phi| \from |\mathcal U| \to |\mathcal Y|$.
If $\phi$ is continuous in one-parameter families, then it is induced by a morphism $\Phi \from \mathcal X \to \mathcal Y$ that extends $\Phi \from \mathcal U \to \mathcal Y$.
\end{lemma}
By \autoref{lem:unique_extension}, the extension is unique.
\begin{proof}
Consider the map $(\id, \Phi) \from \mathcal U \to \mathcal X \times \mathcal Y$.
Let $\mathcal Z \subset \mathcal X \times \mathcal Y$ be the scheme theoretic image of $\mathcal U$ (see \cite[Tag 0CMH]{sta:20}), and let let $\mathcal Z^\nu \to \mathcal Z$ be the normalization.
By construction, the map $\mathcal Z \to \mathcal X$ is an isomorphism over $\mathcal U$ \cite[Tag 0CPW]{sta:20}.
Since $\mathcal U$ is smooth (and hence normal), the map $\mathcal Z^\nu \to \mathcal X$ is also an isomorphism over $\mathcal U$.
Our aim is to show that $\mathcal Z^\nu \to \mathcal X$ is in fact an isomorphism.
Let $Z^\nu \to X$ be the morphism on coarse spaces induced by $\mathcal Z^\nu \to \mathcal X$.
Let $(x, y) \in |\mathcal X| \times |\mathcal Y|$ be a point from $Z^\nu$.
Since the image of $\mathcal U$ is dense in $\mathcal Z^\nu$, there exists a DVR $\Delta$ with a map $\Delta \to \mathcal Z^\nu$ whose generic point maps into the image of $\mathcal U$ and whose special point maps to $(x,y)$.
The continuity of $\phi$ in one-parameter families implies that $y = \phi(x)$.
As a result, $Z^\nu \to X$ is a bijection on points.
As $Z^\nu$ and $X$ are normal spaces, $Z^\nu \to X$ must be an isomorphism.
By hypothesis, for every DVR $\Delta$, a map $\Delta \to \mathcal X$ that sends the generic point to $\mathcal U$ lifts to a map $\Delta \to \mathcal Y$, and hence to a map $\Delta \to \mathcal Z^\nu$.
This implies that $\mathcal Z^\nu \to \mathcal X$ is unramified in codimension 1.
It follows by the same arguments as in \cite[Corollary~6]{ger.sat:17} that $\mathcal Z^\nu \to \mathcal X$ is an isomorphism.
Since \cite[Corollary~6]{ger.sat:17} is stated with slightly stronger hypotheses, we recall the proof.
Let $V$ be a scheme and $V \to \mathcal X$ an \'etale morphism.
Set $\mathcal W = \mathcal Z^\nu \times_{\mathcal X} V$ and $U = \mathcal U \times_{\mathcal X} V$.
Let $\mathcal W \to W$ be the coarse space.
The map $W \to V$ is an isomorphism over the dense open subset $U \subset V$, and is a quasi-finite map between normal spaces.
By Zariski's main theorem, it is an isomorphism.
Furthermore, as $\mathcal W \to V$ is unramified in codimension 1, so is $\mathcal W \to W$.
Since $\mathcal W$ is normal and $W$ is smooth, purity of the branch locus \cite[Tag 0BMB]{sta:20} implies that $\mathcal W \to W$ is \'etale.
As $\mathcal W \to V$ is an isomorphism over $U$, we see that $\mathcal W$ contains a copy of $U$ as a dense open substack.
In particular, $\mathcal W$ has trivial generic stabilizers.
By \cite[Lemma~4]{ger.sat:17}, we conclude that $\mathcal W \to W$ is an isomorphism.
Since both $\mathcal W \to W$ and $W \to V$ are isomorphisms, their composite $\mathcal W \to V$ is an isomorphism.
We have proved that $\mathcal Z^\nu \to \mathcal X$ is an isomorphism \'etale locally on $\mathcal X$.
We conclude that $\mathcal Z^\nu \to \mathcal X$ is an isomorphism.
The composite of the inverse of $\mathcal Z^\nu \to \mathcal X$, the map $\mathcal Z^\nu \to \mathcal X \times \mathcal Y$, and the projection onto $\mathcal Y$ gives the required extension $\Phi \from \mathcal X \to \mathcal Y$.
\end{proof}
\begin{proof}[Proof of \autoref{thm:phi_extends}]
Define a map $\phi \from |\overline{\mathcal {H}}_4^3(1/6+\epsilon)| \to |\mathcal X|$, consistent with the map induced by $\Phi$ on $U$ as follows.
Let $k$ be an algebraically closed field and $a \from \spec k \to \overline{\mathcal{H}}_4^3(1/6 + \epsilon)$ a map.
Let $\Delta$ be a DVR with residue field $k$ and $\alpha \from \Delta \to \overline{\mathcal{H}}_4^3(1/6 + \epsilon)$ a map that restricts to $a$ at the special point and maps the generic point $\eta$ to $U$.
By \autoref{prop:stab}, there exists an extension $\beta \from \Delta \to \mathfrak X$ of $\Phi \circ \alpha |_\eta$.
Let $b \from \spec k \to \mathfrak X$ be the central point of the extension.
Set $\phi(a) = b$.
\autoref{prop:stab} guarantees that $b$ depends only on $a$, and not on $\alpha$; so the map $\phi$ is well-defined.
\autoref{prop:stab} also guarantees that $\phi$ is continuous in one-parameter families.
Since $\overline{\mathcal H}^3_4(1/6 + \epsilon)$ is smooth, \autoref{lem:exists_extension} applies and yields the desired extension.
\end{proof} | 141,173 |
PUBLIC RECORD
312 Illinois St Lawrence, KS 66044 (Pinckney)See your commute times
- Single-Family Home
- 3 Bedrooms
- 2 full Bathrooms
- 900 sqft
- Lot size: 6,098 sqft
- Built in 1950
- Edit Home Facts
Property Details for 312 Illinois St
Description provided by Trulia
This is a Single-Family Home located at 312 Illinois Street, Lawrence, KS. 312 Illinois St has 3 beds, 2 baths, and approximately 900 square feet. The property has a lot size of 1 and was built in 1950. The average list price for similar homes for sale is $188,170. 312 Illinois St is in the Pinckney neighborhood in Lawrence, KS. The average list price for Pinckney is $98,981.
Public Records for 312 Illinois St
Official property, sales, and tax information from county (public) records:
- 3 Bedrooms
- Lot Size: 6,098 sqft
- 2 Bathrooms
- Built In 1950
- 900 sqft
- County: Douglas | 336,665 |
TITLE: Proof of Cancellation Law for Multiplication in N (Induction)
QUESTION [2 upvotes]: I'm struggling a bit to prove the Cancellation Law for multiplication for the Natural numbers, using induction.
So far I have this:
Having $a, b \in N$ e $S = \lbrace c \in N | a \cdot c = b \cdot c \longrightarrow a = b \rbrace$
We know that $1 \in S$, since $a \cdot 1 = a = b = b \cdot 1$, 0 is not the base case because even if $a \neq b$, $a \cdot 0 = 0 = b \cdot 0$.
Using $k \in S$, by definition $a \cdot k = b \cdot k \longrightarrow a=b$, we have to show that $k' \in S$
Assuming that $a \cdot k' = b \cdot k'$, we have $a \cdot k + a = b \cdot k + b$, and I don't know where to go from here.
I tried using something like this, however I'm still stuck.
Thanks for any help
REPLY [0 votes]: There is another approach which requires defining a map:-
Let $x,y\in\mathbb{N}$. Define set $T=\{z\in\mathbb{N}\setminus\{0\}:xz=yz \implies x=y\}$.
Instead of showing that $T=\mathbb{N}\setminus\{0\}$, we define $S=T\cup\{0\}$ and show that $S=N$ by peano's inductive axiom.
Let $s:\mathbb{N}\to \mathbb{N}$ be the successor function with properties as defined in Peano's $5$ axioms characterizing $\mathbb{N}$.
$0\in S$ (by construction).
Assume for $z\in\mathbb{N}$, we have $z\in S$, i.e., $xz=yz \implies x=y$.
Define map $f:\mathbb{N}\to\mathbb{N}$ as $f(n)=n\times s(z)$, where $z$ is a fixed number as assumed above.
It is easy to show that $f$ is injective.
(Inductive step): $x\cdot s(z) = y\cdot s(z)$
$\implies f(x)=f(y)$
$\implies x=y$ ($\because f$ is injective).
Thus, we have shown: $z\in S \implies s(z)\in S$. Hence, $S=\mathbb{N}$. | 107,848 |
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MAZ 5432
Tested: 1.8.2.5s
- 2014-01-18 10:36
- Author and chassis tuning: Bruieser
- Author cabin: Jawa
- Author envelope in ETS2: Dalnoboishik ™ (ie I)
- Authors Salon: Bruiser, Stalker45, Mr.Nick
- Author envelope salon in ETS2: Stalker45
- Reteksturing salon vneshke: Versetti (Gray Tabby) | 43,282 |
The.
[RELATED: Mark Ragan and Jim Ylisela present advanced writing and editing tips for corporate communicators. Join them in Chicago, Washington D.C., or Denver!]
So you're telling me I spent all that time memorizing state abbreviations only to throw them out now?
— Adam Lichtenstein (@ALichtenstein24) April 23, 2014
So you're telling me I spent all that time memorizing state abbreviations only to throw them out now?
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Thank you to our members and guests for attending today’s Educational Short Course on the Valuation Process in Ag Lending.
Thank you to Paige Gilligan and Stan Xavier for their time and for sharing their knowledge and experiences. The presentations were very insightful and tied together very well.
Also, thank you to MetLife Agricultural for allowing Ag Lenders' use of the beautiful conference room.
Past Short CoursesEducational Short Course - August 1, 2019, presented by Paige Gilligan and Stan XavierLegal Short Course - February 26, 2019, presented by Baker Manock & JensenLegal Short Course - October 26, 2018, presented by Steven Turner, Michael Gomez, and Frank OliverAccounting Short Course - September 19, 2018, presented by Baker Peterson Franklin, CPAEducational Short Course - July 12, 2018 - presented by First American TitleLegal Short Course - February 22, 2018 - presented by Bolen Fransen Sawyers Legal Short Course - February 28, 2017 - presented by FrandzelEducational Short Course - July 14, 2017 - presented by Old Republic Title Legal Short Course - 2016 - presented by Bolen Fransen Sawyers LLPAccounting Short Course - 2015 - presented by Baker Peterson Franklin, CPA, LLP
Copyright © 2020 Ag Lenders Society of California | Contact Us | 18,516 |
Day one of Investinblue, (15-16thJune) will include an interactive, industry led, panel debate, highlighting the environmental challenges faced by the maritime industry.
“This House believes that the UK and other countries are meeting the economic resilience requirements to address increasing storms” will be the topical debate closing the first day of the Investinblue conference, taking place during IFB2016.
Peter Green, CEO of National Maritime (NMDG), organisers of Investinblue, said; “The debate will essentially focus on the increasing ratio and force of storms, and whether the UK and other countries may have to re-address the economic resilience factor and necessary infrastructure and innovations that may be required.” The complex pattern of storm damage affecting economic marine activity, inland urban and rural areas and key infrastructure is questioning whether existing resources and capabilities are sufficient. It is likely that if economic activities and growth plans are to be realised, the scale and approach towards such resilience may have to be more transformational.
This presents opportunities for new industry growth and refined flood management, within a climate challenging economy. Recent events have proven the case that despite investment in capital projects, the customer (house owners and businesses), are being negatively impacted by flood.
The cost of mopping up is huge and the process is time consuming. One example of a capital project is that Environment Ministers are going ahead with a five year ‘River Catchment Scheme’, designed to re-direct river flows into natural holding plains before reaching towns or cities.
The cost of repairing the damage spans many sectors and amounts to a significant investment opportunity. Given the complexity of these funding sources, can the case be made for switching the current costs of inundation damage into innovation and improving our use of science and engineering to reduce future inundation risk?
Chaired by Bruce Thomson, Managing Director, Accelerate Corporation, the debate’s participants will include Professor Peter Guthrie, Professor of Engineering for Sustainable Development, University of Cambridge, Professor Sean Smith BSc, PhD, FIOA, FRSA, Professor of Construction Innovation & Director of ISC, Institute for Sustainable Construction, School of Engineering & the Built Environment, Edinburgh Napier University, and also aims to include the day’s key speakers.
For further information and how to book for the event visit:
Further information on sponsorship:
Twitter: @maritimedevgrp @Jobsinmaritime
- Investinblue, a two day event during at IFB2016, a biennial marketplace which brings together thousands of businesses from around the world for three weeks of events, networking and deal-making.. Investinblue will promote UK maritime infrastructure development, high technology projects, products, systems and services to the world, bringing together the UK maritime sector with the goal of working together to showcase, draw interest and inbound investment from UK and overseas stakeholders, dedicated to stimulating the growth of the UK and maritime economy.
-: | 332,988 |
Enroll in Centennial Schools Today!
Ellie Jensen, a seventh grade student at Centennial Middle School, is a semifinalist eligibile to compete in the 2014 Minnesota State Geographic Bee, Friday, April 4.
The North Suburban Cougars finished third in the state tournament. Congrats to team and coaches on a superb 13-1-1 season!
Congratulations to the CHS girls basketball team, Section 5AAAA champions and MSHSL State Girls Basketball Tournament participant.
Odyssey, a school-within-a school, provides full-time instruction delivered in self-contained classrooms for Level Four gifted students in grades three and four. Odyssey begins in the fall of the 2014-2015 school year.
Centennial Schools to Add Full-Time Gifted Program (Star Tribune, 4/1/14)
Congratulations to 2014 All-State Band Members Emily Green and Mackenzie Smith; soloists Lexie Glaser and Emily Green for "Best @ Site" commendations at the Region 5AA Instrumental Solo & Ensemble Contest; and Centennial students receiving 27 "excellent" ratings and 28 "superior" ratings at the contest.
MPR's Music with Minnesotans: Emily Green | 97,425 |
This Man Ate 2 Tbsp. Of Coconut Oil For 60 Days…See What Happened To His Brain
Dr..’
According to Dr. Mary Newport, traditional medicines they were using weren’t working at all and in fact, and were only producing terrible side effects for Steve. She decided that she wanted him to try more natural healing methods and when she did some research, she found out something surprising. | 306,343 |
Monday, June 25, 2018
The new OneIT Strategic Plan details strategic goals OneIT will pursue in support of the institutional mission, and how OneIT will evolve to meet technology needs now and in the future. Drafting the plan was a highly collaborative effort that involved significant engagement of campus stakeholders, IT leaders, and the broader IT community. | 92,937 |
Item #: AB-GR
USOA boot socks are not your average socks. They're designed to excel in harsh conditions to serve military personnel, public safety professionals, outdoor sports enthusiasts and anyone with an active lifestyle. Made from a blend of silver-plated nylon, cotton and spandex, these socks provide a close, comfortable fit. Silver-plated nylon is an antimicrobial fiber that eliminates a broad spectrum of bacteria. It works on contact to eradicate all foot odor. These advanced socks also wick moisture away from the foot, regulate temperature, improve resistance to blisters and provide anti-static features. Plus, they're made in America and come with three pairs in a pack. When you have extraordinary work to do, ordinary socks won't do. USOA socks perform and protect in the most extreme situations.
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Response from WorkBoots.com | 65,908 |
: Business News, National News, News Tags: Companies, Fortune 500, Walmart Send to a friend! Share on Facebook Share on Twitter Pin it! Reddit This! Share on Tumblr! | 45,094 |
5thRound.com will once again be covering UFC 108 tomorrow night. Spike TV will be kicking off the day’s festivities with a live broadcast of two preliminary bouts beginning at 9PM ET/6PM. Unfortunately, you’re gonna have to fork over $44.95 (or 54.95 to have it in HD) for the good stuff starting at 10PPM ET/7PM PT.
The card will be headlined by a light heavyweight scrap between former champ Rashad Evans and the electrifying Thiago Silva. A win is imperative for both if they intend to avenge their respective losses to reigning title holder Lyoto Machida.
Don’t forget that you can stream the UFC 108 post-fight press conference live right here on 5thRound.com at approximately 1AM ET/10PM PT.
Here is a complete breakdown of today’s schedule.
Preliminary Bouts:
Rafaello Oliveira vs. John Gunderson
Jake Ellenberger vs. Mike Pyle
Mark Munoz vs. Ryan Jensen
| 347,764 |
Mayhem 45
The hybrid top and panel opening of this pack makes it ideal for accessibility in tight environments like the local crag, hostels, or backcountry huts. Built with Spectra fiber, the Mayhem delivers maximum strength and abrasion resistance with a light and comfortable carry.
Features/Specifications:
- Front Panel Expansion Pocket With Daisy Chain Attachment Loops And Built In Haul Handle
- Hybrid Panel/ Top Loader System, Ideal For Trail And Travel
- Hex Ridged Eva Backpanel Padding For Omni-Directional Airflow
- Dual Density Eva/Pu Shoulder Straps
- High Density Foam Framesheet With Wireframe Load Dispersion
- Forward Pull Waitbelt With Side Adjustment Webbing
- Ventilated Waistbelt Padding
- Stretch Mesh Waistbelt Pocket
- Top Lid Zippered Compartment With Secondary Map Pocket Underneath
- Side Panel Mesh Bottle Pockets With Over/Under Strap Pass Thru
- Dual Tool/Trekking Pole Carry With Elastic Pole Tip Loops
- Ice And Mixed Tool Lower Panel Storage Sleeves
Specifications:
- Dimensions: 22X10.5X7 (Hxwxd)
- Torso Size: 15-18
- Waistbelt: 28-48
- Weight: 2 Lbs 9 Oz / 1.2 Kg
- Volume: 29 L
- Capacity: Up To 40 Lbs | 23,209 |
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