TITLE: Chomsky normal form method: CYK parser performance implications?
QUESTION [9 upvotes]: Chart parsers can be implemented based on Chomsky normal form or
directly based on production rules. Lets for the moment assume
we have a CYK chart parser that uses Chomsky normal form. The
binarization is not uniquely defined. Does this impact the
performance of the CYK chart parse. Can this be exploited to
improve the performance of a CYK chart parser?

REPLY [4 votes]: Actually, Chomsky normal form (CNF) is not need to run CYK, only
binarization. Binarization is essential to preserve cubic complexity
of parsing, though essential only with respect to non-terminals
(NT). But then, if you have rules including only 2 non-terminals and
some terminals, the CYK algorithm becomes more complex to program and
explain.
As you say, there are many ways to do binarisation. Some will yield
smaller grammars than other. For example
X -> B C D
Y -> B C E

can be binarized as
X -> Z D
Y -> Z E
Z -> B C

thus saving one rule by factorization, which may save on the computation, and on its result size.
But with other rules, you might want to factorize the end of the rules
rather than the beginning.
I am not familiar with the work of Song, Ding, and Lin, cited by Rob
Simmons' answer. The idea is interesting but I wonder how effective it
can be compared to other ways of optimizing the computation. I fear
not so much.
The point is that analyzing the issues only with respect to a pure CKY
algorithm seem a bit an academic but costly exercise since there are
other kinds of optimization that can significantly improve the
elimination of dead end parses.
CYK is only one of the simpler variations in a family of algorithms
that are all built on the same dynamic programming model, apparently.
I am saying apparently because the simplest version of these algorithms
is not known as dynamic programming, but as cross-product. It is the
old construction of a CF grammar G that generates the intersection of the
language of CF grammar F and the regular language of a FSA A., due to
Bar Hillel, Perles and Shamir (1961), as remarked by Lang in 1995.
All chart parser, or general CF parsers based on dynamic
programming may be seen as "optimized" variant of that cross-product
construction, the optimization being used mainly to avoid useless
computations of the parser. But the problem is subtle as avoiding
useless computation may result in duplicating useful ones, which may be
worse.
Being bottom-up, the CKY algorithm produces useless computations of
partial parses that cannot derive from the axiom of the grammar.
Algorithms like the GLR parser (to name one of the better known ones, though flawed version have been published), have some top-down knowledge that will avoid many
such useless computations, possibly at cost. And there are many other
variants with different behavior with respect to saving on useless computations..
It is with these optimization strategies in mind that the binarization
strategy should be analyzed. What is the point of optimizing what may
be a minor issue, and ignore more powerful techniques.
The optimization of the parsing process is also tightly linked to the
"quality" of the parse-structure obtained, that represent all
possibles parses, and is often called (shared-)parse-forest. I discuss that in another answer.
Some of these issues are discussed in the literature. For example by Billot and Lang analyse some aspects of binarisation with respect to parsing strategies.


TITLE: Hypergeometric sum expressed through elliptic integrals
QUESTION [2 upvotes]: Let $n\ge 0$ be an integer and let $m\in(0,1)$ be a real number. Consider a following sum:
\begin{equation}
{\mathfrak I}^{(n)}_m := \frac{\pi}{2} (-1)^n \sum\limits_{j=0}^\infty \binom{-1/2}{j} \binom{-1/2}{j+n} m^j
=\frac{\pi}{2} (-1)^n\binom{-\frac{1}{2}}{n} \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+1;m\right).
\end{equation}
It is not hard to see that this sum has a following integral representation:
\begin{equation}
{\mathfrak I}^{(n)}_m = \int\limits_0^{\pi/2} \frac{\cos(u)^{2 n}}{\sqrt{1-m \cos(u)^2}} du = \int\limits_0^1 \frac{y^{2 n}}{\sqrt{1-m y^2}}\cdot \frac{1}{\sqrt{1-y^2}}dy.
\end{equation}
The integrals above  can be always reduced to elliptic integrals by using the procedure in http://mathworld.wolfram.com/EllipticIntegral.html . We have actually done this and then we found that the quantities above satisfy the following recurrence relations:
\begin{eqnarray}
{\mathfrak I}^{(n)}_m - \sum\limits_{q=0}^{n-1}(1+m^{q-n}) {\mathfrak I}^{(q)}_m = \frac{\sqrt{m} E\left(\sin ^{-1}\left(\sqrt{m}\right)|\frac{1}{m}\right)-m^{-n} E(\pi/2|m)}{1-m}
\end{eqnarray}
where $E(\phi|m)$ is the elliptic integral of the second kind.
Now, my question is the usual one. Is it possible to find a closed form expression for the quantities in question?

REPLY [2 votes]: I agree with the comment of reuns, but I also would like to point out that $m=\frac{1}{2}$ is one of such special values. In such a case
$$\mathfrak{I}^{(n)}=\int_{0}^{\pi/2}\frac{\sin(\theta)^{2n}\,d\theta}{\sqrt{1-\frac{1}{2}\sin^2\theta}}=\frac{\pi}{2}\sum_{h\geq 0}\frac{\binom{2h}{h}\binom{2h+2n}{h+n}}{16^h\cdot 4^{h+n}} $$
is always given by a linear combination with rational coefficients of $E\left(\frac{1}{2}\right)$ and $K\left(\frac{1}{2}\right)$, where
$$ K\left(\tfrac{1}{2}\right) = \tfrac{1}{4\sqrt{\pi}}\,\Gamma\left(\tfrac{1}{4}\right)^2,\qquad E\left(\tfrac{1}{2}\right) = \frac{\pi\sqrt{\pi}}{\Gamma\left(\frac{1}{4}\right)^2}+\frac{\Gamma\left(\frac{1}{4}\right)^2}{8\sqrt{\pi}} $$
by Clausen's formula (${}_2 F_1(\ldots)^2 = {}_3 F_2(\ldots)$) and Legendre's identity.


TITLE: 11 sided irregular shape that tessellates
QUESTION [2 upvotes]: My friend was fiddling around on the triangle when he created an irregular heptagon with it not able to tessellate.

He then asked me if I could create an 11 sided irregular polygon that is able to tessellate by changing a side AB. He told me there was 3 and that you can make 2 more without necessarily using the side AB. I have absolutely no idea what they are. Can I please have some help?

REPLY [1 votes]: For: "create an 11 sided irregular polygon that is able to tessellate by changing a side AB":

which then becomes:

where I just a projected a duplicate triangle, using the axis of symmetry between side AB (to be changed) and side AC (the side with tessellation to be matched with). And there you have your shape. There are more you can make, take this example as a hint.
For: "you can make 2 more without necessarily using the side AB":
Here's another hint:

Good Luck!


TITLE: Expected Value where probability changes after success
QUESTION [0 upvotes]: After 1 success of a series of trials, the probability of success increases from 1/2 to 21/36. How do I calculate the expected number of trials until I reach 5 successes?
Naively, it seems to be a geometric distribution and a negative binomial distribution: I expect the first 2 trials to give 1 success, and I expect 7 trials to give just over 4 successes afterwards, so 9 trials leads to 5 successes.
To get a better answer, it seems I should calculate the probability of getting the first success on the n-th trial, then multiply by a binomial distribution of some sort.
However, there is a (very small) chance of still not getting a single success in n+1 trials, and 4 more successes in arbitrarily many trials, and I don't see how to integrate across the two infinities.

REPLY [0 votes]: The number of trials from 0 successes to 1 success, 1 success to 2 successes, etc. is geometrically distributed, with expected value 1/p.
Therefore, the expected number of trials for the 1st success is 1/(.5) = 2
The expected number of trials from the 1st to the 2nd success (and from 2nd to 3rd, etc.) is 1/(21/36) = 36/21 ~ 1.71
Therefore, the expected number of trials is 186/21 ~ 8.85


TITLE: Is there any construction of real numbers that does not use a quotient space in the process?
QUESTION [4 upvotes]: I am working on an isomorphism (in terms of order and operations) from the power set of integers to R. I would like to know if anyone knows of any construction of the real numbers that uses such a simple set to provide it with the structure of the real number system. Thanks!

REPLY [5 votes]: If we take the rationals as given (or construct them without using a quotient procedure - see my comment below), then the Dedekind cut construction, I think, does what you want.
A real number, the idea goes,$^*$ is completely characterized by the way it "cuts" the number line into two pieces. This cut is completely determined by which rationals wind up on which side, and conversely every "reasonable" partition of the rationals corresponds to a real number (this is the completeness of the reals). This motivates the following definition:

A Dedekind cut is a pair $(A, B)$ of nonempty sets of rationals such that every element of $A$ is less than every element of $B$, $A$ has no greatest element, and $A\cup B=\mathbb{Q}$.

The intuition here is that $(A, B$) is the unique real number "between" $A$ and $B$. E.g. the cut $(A_\pi, B_\pi)$ corresponding to $\pi$ has $3, 3.1, 3.14, ...\in A_\pi$ and $4, 3.2, 3.15, ...\in B_\pi$.
The asymmetry between $A$ and $B$ - where $A$ must have no greatest element, but $B$ is allowed to have a least element - is to accommodate the rationals. The cut corresponding to ${1\over 2}$ needs to put ${1\over 2}$ on one side or the other; we've arbitrarily decided that in this situation we'll put ${1\over 2}$ on the right (the $B$-side).
There is no need to take equivalence classes in this approach: given any real number $r$, there is a unique Dedekind cut corresponding to it, namely $$Cut_r=(\{p\in\mathbb{Q}: p<r\},\{q\in\mathbb{Q}: q\ge r\}).$$ The basic arithmetic structure on the reals "ports over" without serious difficulty; for example, the ordering is given by $$(A_1, B_1)\le (A_2, B_2)\iff A_1\subseteq A_2.$$ That said, the Cauchy sequence construction of the reals allows a much easier development of arithmetic; my point here is just that no serious difficulty crops up with cuts.

Note that the notion of a Dedekind cut, while geometrically pleasing, is somewhat redundant. We could also just work with the "right side" of a cut:

A half-cut (this is not standard terminology) $C$ is a nonempty upwards-closed proper subset of $\mathbb{Q}$.

The idea is that the real associated to $C$ is its greatest lower bound.

$^*$Personally, I think Dedekind's original essay does a wonderful job of motivating the idea, and it's still my favorite construction of the reals. The construction of the reals as equivalence classes of Cauchy sequences has lots of advantages over it - in particular, defining the basic arithmetic operations is easier with them than with cuts - but I really find Dedekind cuts to have lots of aesthetic value. And down the road, they wind up being quite important and useful in logic.


TITLE: The Bruss-Yor conjecture about an iterated integral
QUESTION [17 upvotes]: Is the sequence $$w_n=n! \int_0^{1/2} \int_{x_1}^{2/3} \cdots\int_{x_{n-2}}^{\frac{n-1}{n}} \int_{\frac{n}{n+1}}^1  dx_n dx_{n-1} \cdots dx_1$$ increasing for $n\ge 3$?

This is a conjecture of F. Thomas Bruss and Marc Yor in a recent paper Stochastic Processes with Proportional Increments and The Last-Arrival Problem. It is presented here with the permission of them.
Background
In the mentioned paper Bruss and Yor introduce stochastic processes with proportional increments to deal with the so called last arrival problem (l.a.p.) where an unknown number of independent random random variables $X_1,\ldots,X_N$ (which are uniformly distributed on the interval $[0,1]$) is observed and "our objective is to
stop online with exactly one stop on the very last of these points, i.e. at their
largest order statistics $X_{\langle N;N\rangle}$" (cited from Bruss and Yor).
Bruss and Yor give an interpretation of the problem (it is not so obvious what it means that $N$ is unknown) and then provide the optimal strategy. The numbers $w_n$ are then
the win probability conditioned on $N=n$. Bruss and Yor state that $w_n \to 1/e$.
Some relations
The inner integral clearly is $1/(n+1)$ and the remaining integral is thus up to the factor $n/(n+1)$ the probability that the order statistic $X_{\langle 1;n-1\rangle},\ldots,X_{\langle n-1;n-1\rangle}$ of $n-1$ independent on $[0,1]$ uniformly distributed random variables has values in $[0,1/2] \times [0,2/3] \times \cdots \times [0,\frac{n-1}{n}]$. It would be thus natural if a similar problem had appeared in some statistical context.
One can write down the latter probability using multinomial sums (which looks very
ugly). Nevertheless this suggests that some clever combinatorial arguments might help.
Finally the order statistics are closely related to Poisson processes (however, as the authors are experts in stochastic processes they have most probably checked that area).

REPLY [9 votes]: Here is a sketch of a full proof that can be (at least, in principle) verified by a human.
Claim 1. $\gamma_n=(\frac{n-1}{n})^{n-1}$ is a decreasing sequence.
This is well known and easy to prove.
Claim 2. The integral in question is just $\frac n{n+1}$ times the probability $P_n$ that $Y_k\le \frac k{k+1}$ for all $k=1,\dots,n-1$ where $Y_k$ is the $k$-th value in the list obtained by sampling $n-1$ independent random variables uniformly distributed on $[0,1]$ and putting them in the increasing order.
This has already be mentioned by the OP.
Claim 3. The joint distribution of $Y_{n-1},\dots,Y_1$ is the same as of $X_1^{\frac 1{n-1}},X_1^{\frac 1{n-1}}X_2^{\frac 1{n-2}},\dots,X_1^{\frac 1{n-1}}X_2^{\frac 1{n-2}}\cdots X_{n-1}$ where $X_j$ are i.i.d. random variables uniformly distributed on $[0,1]$.
Just condition on the maximum and use induction. From now on, we will just assume that $Y_k=X_1^{\frac 1{n-1}}\dots X_{n-k}^{\frac 1{k}}$.
Claim 4. For all $k\ge 2$, we have
$$
(\frac{k-1}{k})^{\frac{k-1}k}(\frac{k}{k+1})^{\frac 1{k(k+1)}}\le \frac{k}{k+1}
$$
Rewrite it as $(\frac{k^2-1}{k^2})^{k^2-1}\le (\frac{k}{k+1})^{k}$ and use Claim 1.
Claim 5. Let $2\le k\le n-2$. If $X_1^{\frac 1{n-2}}\dots X_{n-k}^{\frac 1{k-1}}\le \frac{k-1}{k}$ and $X_1^{\frac 1{n-2}}\dots X_{n-k-1}^{\frac 1{k}}\le \frac{k}{k+1}$, then
$X_1^{\frac 1{n-1}}\dots X_{n-k}^{\frac 1{k}}\le \frac{k}{k+1}$.
Raise the first inequality to the power $\frac{k-1}{k}$, the second to the power $\frac 1{k(k+1}$, multiply, use the fact that $X^t$ is decreasing in $t$ for $0<X<1$, and apply Claim 4.
Claim 6.
$$
P_n\ge \frac{\gamma_n}{\gamma_{n-1}}P_{n-1}-(1-\frac 2n)^{n-1}2^{-(n-1)}
$$
Claim 5 allows one to conclude that replacing a sample $X_1,X_2,\dots$ satisfying the desired set of inequalities for $n-1$ instead of $n$ by $X_1'=\frac{\gamma_n}{\gamma_{n-1}}X_1, X_2'=X_2, X_3'=X_3,\dots$, we get a sample satisfying all the desired inequalities for $n$ except, perhaps, the very last one. Thus, the probability $Q_n$ that all inequalities except the last one are satisfied is at least $\frac{\gamma_n}{\gamma_{n-1}}P_{n-1}$ (the factor comes from the bound for the density of $X_1'$ relative to the density of $X_1$). What we need to subtract is at most the probability that the minimum of $n-1$ independent random variables uniformly distributed on $[0,1]$ is at least $1/2$ while their maximum is at most $\frac{n-1}n$. This is the last term in the inequality.
Claim 7. We have
$$
\frac{n}{n+1}P_n\ge \frac{n}{n+1}\frac{n}{n-1}\frac{\gamma_n}{\gamma_{n-1}}\frac{n-1}n P_{n-1}-\frac{n}{n+1}(1-\frac 2n)^{n-1}2^{-(n-1)}
$$
This is just Claim 6 rewritten.
Assuming that $\frac{n-1}nP_{n-1}\ge\frac 5{16}$, we conclude that we will be able to run the induction if
$$
[\frac{n}{n+1}(1+\frac{1}{n(n-2)})^{n-2}-1]*\frac{5}{16}\ge \frac{n}{n+1}(1-\frac 2n)^{n-1}2^{-(n-1)}
$$
Using the first three terms in the Newton's formula on the left, we see that to pass from $n-1$ to $n$, it is enough to have
$$
\frac{5}{16}\frac{n-3}{2n^2(n-2)}\ge (1-\frac 2n)^{n-1}2^{-(n-1)}
$$
Claim 8. This inequality holds for $n\ge 7$.
For $n=7$, it reduces to $8\cdot 7^4>5^6$, i.e., $(2-\frac 1{25})^2>4-\frac 78$, which is fairly obvious by Bernoulli. It remains to note that when $n$ goes up by $1$, the left hand side is multiplied by at least $(\frac 78)^2$\ge $\frac 34$ while the right hand side is multiplied by $1/2$ because of the power of $2$ and at most $(1-\frac{n-1}{n(n-2})^{-1}\le \frac{35}{29}$, which is the Bernoulli bound for the factor by which $(1-\frac 2n)^{n-1}$ differs from the decreasing sequence $(\frac{n-1}{n})^2(n-1)$. Now, $\frac 12\cdot\frac {35}{29}\le \frac{18}{29}\le \frac 34$, so we can run the induction.
The most difficult (=time and effort consuming) part of this proof is
Claim 9. $P_1$ through $P_6$ are fine for the base of induction.
I managed to compute $P_4$ by hand without an arithmetic error, but not $P_5$. That's why I would still prefer to see an argument that doesn't require a direct computation of any value beyond $P_3$.


TITLE: Basic Geometry Sum
QUESTION [0 upvotes]: Suppose P,Q,R and S are the midpoints of the sides AB,BC,CD and DA, respectively of a rectangle ABCD. If the area of the rectangle is delta, then area of the figure bounded by straight lines AQ,BR,CS and DP is?
Please help.Thanks.

REPLY [1 votes]: Let $AQ\cap DP=\{L\}$, $AQ\cap BR=\{M\}$, $CS\cap DP=\{K\}$ and $CS\cap BR=\{N\}$.
Since $AQCS$ and $DPBR$ are parallelograms, we see that $KLMN$ is parallelogram.
Now, let $F$ is a middle point of $RC$.
Thus, $QF||BR$, which says that $G$ is a middle point of $NC$, where $\{G\}=QF\cap NC$.
In another hand, since $R$ is a middle point of $DC$ and $RB||DP$, we obtain that $N$ is a middle point of $KC$.
By the same way we can get that $AL=LM=KN=NC$ and $DK=KL=BM=MN$
and since $AQ=CS$ and $DP=BR$, we obtain $SK=MR$ and $PL=RN$.
Now, since $NG=GC$ and $MNGQ$ is parallelogram we obtain that $S_{\Delta GQC}=\frac{1}{3}S_{MNCQ}.$
But $GC=GN=MN=SK$, which says that $GC=\frac{1}{5}CS$ and $S_{\Delta GQC}=\frac{1}{10}S_{AQCS}$.
Hence, $S_{MQCN}=\frac{3}{10}S_{AQCS}$.
By the same way we can get that $S_{ALKS}=\frac{3}{10}S_{AQCS}$, which says that
$$S_{KLMN}=S_{AQCS}-2\cdot\frac{3}{10}S_{AQCS}=\frac{2}{5}S_{AQCS}=\frac{\Delta}{5}$$
and we are done!


TITLE: Inequality. $\sum{\sqrt{x^2+xy+y^2}}\geq \sum{\sqrt{2x^2+xy}}.$
QUESTION [0 upvotes]: Let $x,y,z >0$. Prove that:
$$\sum_{\text{cyc}}{\sqrt{x^2+xy+y^2}}\geq \sum_{\text{cyc}}{\sqrt{2x^2+xy}} .$$
Thanks for your help :)

REPLY [0 votes]: We need to prove that
$$\sum_{cyc}\left(\sqrt{x^2+xy+y^2}-\sqrt{2x^2+xy}\right)\geq0$$ or
$$\sum_{cyc}\frac{y^2-x^2}{\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}}\geq0$$ or
$$\sum_{cyc}\left(\frac{y^2-x^2}{\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}}+\frac{x-y}{\sqrt3}\right)\geq0$$ or
$$\sum_{cyc}\frac{(x-y)\left(\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}-\sqrt3(x+y)\right)}{\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}}\geq0$$ or
$$\sum_{cyc}\frac{(x-y)\left(\sqrt{(x^2+xy+y^2)(2x^2+xy)}-y^2-2xy\right)}{\left(\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}\right)\left(\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}+\sqrt3(x+y)\right)}\geq0$$ or
$$\sum_{cyc}\tfrac{(x-y)^2(2x^3+5x^2y+4xy^2+y^3)}{\left(\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}\right)\left(\sqrt{x^2+xy+y^2}+\sqrt{2x^2+xy}+\sqrt3(x+y)\right)\left(\sqrt{(x^2+xy+y^2)(2x^2+xy)}+y^2+2xy\right)}\geq0.$$
Done!


text:
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\vskip -1cm

\title[Exotic Heat PDE's]{\mbox{}\\[1cm] EXOTIC HEAT PDE's}
\author{Agostino Pr\'astaro}

\maketitle

\vspace{-.5cm}
{\footnotesize
\begin{center}
Department of Methods and Mathematical Models for Applied
Sciences, University of Rome ''La Sapienza'', Via A.Scarpa 16,
00161 Rome, Italy. \\
E-mail: {\tt Prastaro@dmmm.uniroma1.it}
\end{center}
}
\vspace{1cm}
\centerline{\it This paper is dedicated to Stephen Smale in occasion of his 80th birthday}

{\footnotesize
\vspace{1cm} {\bf ABSTRACT.} Exotic heat equations that allow to prove the Poincar\'e conjecture, some related problems and suitable generalizations too are considered. The methodology used is the PDE's algebraic topology, introduced by A. Pr\'astaro in the geometry of PDE's, in order to characterize global solutions.\footnote{This paper has been published in \cite{PRA16}.\\Work partially supported by Italian grants MURST ''PDE's Geometry and Applications''.}

\vskip 0.5cm

{\bf AMS (MOS) MS CLASSIFICATION. 55N22, 58J32, 57R20; 20H15.}
{\rm KEY WORDS AND PHRASES. Integral bordisms in PDE's;
Existence of local and global solutions in PDE's; Conservation laws;
Crystallographic groups; Singular PDE's; singular MHD-PDE's.}
}

\section{\bf Introduction}

{\footnotesize\hfill\rightline{\it ''In a category\hskip 5pt $\mathcal{C}$ of manifolds}

\hfill\rightline{\it every homotopy sphere is homeomorphic to a sphere.''}

\begin{figure}[h]
\scalebox{0.9}{$\centerline{\includegraphics[height=2.5cm]{smale-paradox-a.eps}
\includegraphics[height=2.5cm]{smale-paradox-b.eps} \includegraphics[height=2.5cm]{smale-paradox-c.eps}
\includegraphics[height=2.5cm]{smale-paradox-d.eps}}$}
\caption{Smale's paradox: turning a sphere $S^2\subset\mathbb{R}^3$ inside out.}
\label{smale-paradox}
\end{figure}
}
\vskip 1cm

The famous Poincar\'e's conjecture is about $n$-dimensional manifolds, with $n=3$, (R. S. Hamilton \cite{HAMIL1, HAMIL2, HAMIL3, HAMIL4, HAMIL5}, G. Perelman \cite{PER1, PER2}, A. Pr\'astaro \cite{PRA14, AG-PRA1}), but there are also generalizations of this conjecture for higher dimension manifolds. For $n=4$ the Poincar\'e conjecture has been proved by M. Freedman \cite{FREEDMAN, FREEDMAN-QUINN}) and for $n\ge 5$, by S. Smale \cite{SMALE1, SMALE2}. More recently has been given a generalization for quantum supermanifolds by A. Pr\'astaro, that has also proved it in \cite{PRA12}). Nowadays, one can state that a generalized Poincar\'e conjecture can be proved, or disproved, depending on the particular category $\mathcal{C}$ in which it is formulated. This problem aroses in the framework of the geometric topology, but in order to be solved it was necessary to go outside that framework and recast the problem in a theory of PDE's. But the more recent results by A. Pr\'astaro \cite{PRA12, PRA14, AG-PRA1}), have proved that it was necessary to return inside the algebraic topology framework, applied to the PDE's geometric theory. Really, it was soon evident that remaining in a pure algebraic topologic approach it was not enough to solve this conjecture. In fact, a fundamental idea to solve this problem is to ask whether it is possible find a smooth manifold, $V$, that without singular points bords a $3$-dimensional compact, closed, smooth, simply connected manifold $N$ with $S^3$, when $N$ is  homotopy equivalent to $S^3$. The bordism theory is able to state that a smooth manifold $V$ such that $\partial V=N\bigcup S^3$, there exists, since the nonoriented and oriented $3$-dimensional bordism groups $\Omega_3$ and ${}^+\Omega_3$ respectively, are both trivial: $\Omega_3={}^+\Omega_3=0$. However, by simply looking to the above bordism groups it is impossible to state if $V$ has singular points (i.e., has holes) or it is a cylinder. By the way, more informations can be obtained by the {\em h-cobordism theory}. More precisely the {\em h-cobordism theorem} in a category $\mathcal{C}$ of manifolds, states that if the compact manifold $V$ has $\partial V=N_0\sqcup N_1$, such that the inclusion maps $N_i\hookrightarrow V$, $i=0,1$, are homotopy equivalences, (i.e., $V$ is a {\em h-cobordism}), and $\pi_1(N_i)=0$, then $V\cong_{\mathcal{C}} N_0\times[0,1]$. This theorem holds for $n\ge 5$ in the category of smooth manifolds (S. Smale) and for $n=4$ in the category of topological manifolds (M. Freedman). But it does not work for $n=3$!

A very important angular stone, in the long history about the solution of the Poincar\'e conjecture, has been the introduction, by R. S. Hamilton, of a new approach recasting the problem in to solving a PDE, the {\em Ricci flow equation}, and asking for nonsingular solutions there, that starting from a Riemannian manifold $(N,\gamma)$ arrive to the $3$-dimensional sphere $S^3$, respectively identified with initial and final Cauchy manifolds in the Ricci flow equation. In that occasion the Mathematical Analysis, or more precisely the Functional Analysis, entered in the Poincar\'e conjecture problem. This approach has had many improvements until the papers by G. Perelman.
More recently, A. Pr\'astaro, by using his algebraic topologic theory of PDE's was able to give a pure geometric proof of the Poincar\'e conjecture. Let us emphasize that the usual geometric methods for PDE's (Spencer, Cartan), were able to formulate for nonlinear PDE's, local existence theorems only, until the introduction, by A. Pr\'astaro, of the algebraic topologic methods in the PDE's geometric theory. These give suitable tools to calculate integral bordism groups in PDE's, and to characterize global solutions. Then, on the ground of integral bordism groups, a new geometric theory of stability for PDE's and solutions of PDE's has been built. These general methodologies allowed to A. Pr\'astaro to solve fundamental mathematical problems too, other than the Poincar\'e conjecture and some of its generalizations, like characterization of global smooth solutions for the Navier-Stokes equation and global smooth solutions with mass-gap for the quantum Yang-Mills superequation. (See \cite{PRA3, PRA4, PRA5, PRA6, PRA7, PRA8, PRA9, PRA10, PRA11, PRA12, PRA13, PRA15}.\footnote{See also Refs. \cite{AG-PRA1, AG-PRA2, PRA-RAS}, where interesting related applications of the PDE's Algebraic Topology are given.})

The main purpose of this paper is to emphasize some problems related to {\em exotic heat PDE's} recently focused.\footnote{The Ricci flow equation can be considered a generalization of the classical Fourier's heat equation $u_t-\kappa u_{xx}=0$. In this paper we call {\em exotic heat equations}, PDE's that, like the Ricci flow equation, are of the type $F^j\equiv u^j_t-f^j(u^i_k)=0$, $1\le i,j\le m$, with the length $|k|$ of the multi-index $k\in\{1,\cdots,n\}$, given by  $0\le |k|\le s\le r$, where $F^j:J^r(W)\to \mathbb{R}$ are analytic functions of order $r\ge 0$ on a fiber bundle $\pi:W\equiv \mathbb{R}\times E(M)\to \mathbb{R}\times M$, where $E(M)$ is a vector bundle over $M$, with $M$ an analytic manifold of dimension $n$. Let us emphasize that the structure of exotic heat equation is the more suitable to use in order to prove (generalized) Poincar\'e conjectures. In fact, the idea to use PDE's to solve the Poincar\'e's conjecture, was the initial motivation to introduce and study the well-known {\em Yamabe equation} \cite{YAMABE}. But that road did not turn out a lucky choice to prove the conjecture, even if the Yambe equation is a very important equation to study conformal problems in Riemannian geometry. (See, e.g., \cite{AG-PRA1}.)}

Let us conclude this Introduction with a theorem, direct issue from results contained in \cite{PRA14}.
\begin{theorem}\label{homeomorphic-three-manifold-diffeomorphic-three-sphere}
Any $3$-dimensional compact, closed simply connected smooth manifold, $M$, homotopy equivalent to $S^3$, is diffeomorphic to $S^3$.\footnote{This last result agrees with the Hauptvermuntung conjecture that was proved for $(n=2,3)$-dimensional manifolds and disproved for $(n\ge 4)$-dimensional manifolds. (See \cite{CASSON, MILNOR, MOISE1, MOISE2, RADO, SULLIVAN, TUSCH}.)}
\end{theorem}

\begin{proof}
In fact, the method followed by A. Pr\'astaro to prove the Poincar\'e conjecture \cite{PRA14, AG-PRA1}, allows us to conclude that in dimension $n=3$, under the hypotheses of the theorem, one can state that the manifold $V$, smooth solution of the Ricci flow equation, $(\partial t.g_{ij})-\kappa R_{ij}(g)=0$, such that $\partial V=M\sqcup S^3$, has the following structure $V\cong M\times[0,1]$, hence must necessarily be $M\cong S^3$, (in the category of smooth manifolds). In other words the algebraic topology of PDE's, introduced by A. Pr\'astaro, allows us to complete h-cobordism theorem in the category of $3$ dimensional smooth manifolds.
\end{proof}

\section{\bf RICCI FLOW EQUATION IN NON-COMPACT CASE}

In this section we consider the problem of existence and uniqueness of smooth complete solutions of the Ricci flow equation, in the noncompact case. In particular we answer to a recent question put for the $n$-dimensional Euclidean case in \cite{CHEN}. Let us consider the Ricci flow equation, written in the form

\begin{equation}\label{ricci-flow-equation}
\fbox{$(\partial t.g_{ij})(x,t)=-2\, R_{ij}(x,t),\quad (x,t)\in M\times\mathbb{R}$}
\end{equation}

when $M$ is a $n$-dimensional smooth, noncompact and complete Riemannian manifold, $(M,\gamma)$.  In particular the assumption that $M$ is not compact and complete, is a substantial difference with respect to the usual way in which such equation has been considered. In fact, the Ricci flow equation has been principally introduced by R. S. Hamilton, to prove the Poincar\'e conjecture on compact closed $3$-dimensional manifolds homotopic equivalent to $S^3$. (See \cite{{CHOW-CHU-GLIE-GUE-ISE-IVEY-KNOP-LU-LUO-NI-1}, {CHOW-CHU-GLIE-GUE-ISE-IVEY-KNOP-LU-LUO-NI-2}, HAMIL1, HAMIL2, HAMIL3, HAMIL4, HAMIL5, PER1, PER2} to follows the proof of the Poincar\'e conjecture in the approach  by Hamilton-Perelmann. Furthermore, in \cite{PRA14, AG-PRA1} the Poincar\'e conjecture has been proved by using a method different by the Hamilton-Perelmann's one.)

The main motivations to write this section are some recent works where the Ricci flow equation is implemented on a noncompact complete Riemannian manifold, and solutions are request to identify complete metrics on the sectional noncompact manifolds. (See, e.g., \cite{CHEN, SHI}.) More specifically, it was very intriguing the recent paper by B.-L. Chen \cite{CHEN}, where it is emphasized the uniqueness of such solutions in the dimensional $3$, and the following question: ''{\em Does the strong uniqueness of the Ricci flow hold on the Euclidean space $\mathbb{R}^n$, for $n\ge 4$\hskip 2pt ?}\hskip 3pt''.

In fact we have the following theorem.
\begin{theorem}\label{complete-noncompact-case-ricci-flow-equation}
Let us consider the Ricci flow equation {\em(\ref{ricci-flow-equation})}
on the Euclidean space $(\mathbb{R}^n,\gamma)$. Then, the unique solution of {\em(\ref{ricci-flow-equation})} for complete metrics is just $\gamma$.
\end{theorem}

\begin{proof}
Let us also underline that the restriction to consider noncompact complete metrics does not permit flows with contractions, and neither simple homeomorphic ones. Isometric diffeomorphisms are instead permitted. Therefore, let us add to the equation (\ref{ricci-flow-equation}) the condition that $g(x,t)$ is realized by means of a one parameter set of diffeomorphisms $\phi_t:M\to M$, by means of pull-back, i.e, put $g(x,t)=\phi_t^*\gamma$. One has for the Ricci tensor $R_{ij}(x)$ of $\gamma(x)$, the following induced deformation $R(x,t)=R(g(x,t))=\phi_t^*R(\gamma(x))$. In fact, for the natural covariance of the Riemannian metric and its Ricci tensor, one has the following commutative diagram

\begin{equation}
\scalebox{0.7}{$\xymatrix@C=80pt@R=80pt{M\ar@/^3pc/[rrr]_{R(g(x,t))}\ar@/^2pc/[rr]_{D^2g(x,t)}\ar@/^1pc/[r]_{g(x,t)}\ar[d]_{\phi_\lambda}&S^0_2M\ar[l]&J{\it D}^2(S^0_2M)\ar[l]\ar[r]_{R}&S^0_2M\\
M\ar@/_3pc/[rrr]^{R(\gamma(x))}\ar@/_2pc/[rr]^{D^2\gamma(x)}\ar@/_1pc/[r]^{\gamma}&S^0_2M\ar[u]^{S^0_2(\phi_\lambda)}\ar[l]&J{\it D}^2(S^0_2M)\ar[u]^{J{\it D}^2(\phi_\lambda)}\ar[l]\ar[r]^{R}&S^0_2M\ar[u]_{S^0_2(\phi_\lambda)}}$}
\end{equation}

Taking into account that in coordinates $(x^i)_{1\le i\le n}$ on $M$, one has
\begin{equation}\label{ricci-flow-equation-a}
\left\{\begin{array}{l}
         g_{ij}(x,t)=(\partial x_i.\phi^a_t)(\partial x_j.\phi^b_t)\gamma_{ab}(\phi_t(x))\equiv
         (\phi_t)^a_i(x)(\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))\\
          (\partial t.g_{ij})(x,t)=(\dot\phi_t)^a_i(x)(\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))
          +(\phi_t)^a_i(x)(\dot\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))\\
          \hskip 2cm+(\phi_t)^a_i(x)(\phi_t)^b_j(x)(\dot\phi_t)^\alpha(x)(\partial x_\alpha.\gamma_{ab})(\phi_t(x))\\
          R_{ij}(x,t)=(\phi_t)^a_i(x)(\phi_t)^b_j(x)R_{ab}(\phi_t(x))\\
       \end{array}\right.
\end{equation}
we get that equation (\ref{ricci-flow-equation}) can be written as follows

\begin{equation}\label{ricci-flow-equation-b}
{}_\bullet(RF)\subset J{\it D}(W):\hskip 2pt\left\{\begin{array}{l}
          (\dot\phi_t)^a_i(x)(\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))
         +(\phi_t)^a_i(x)(\dot\phi_t)^b_j(x)\gamma_{ab}(\phi_t(x))\\
          \hskip 2cm+(\phi_t)^a_i(x)(\phi_t)^b_j(x)(\dot\phi_t)^\alpha(x)(\partial x_\alpha.\gamma_{ab})(\phi_t(x))\\
          =-2(\phi_t)^a_i(x)(\phi_t)^b_j(x)R_{ab}(\phi_t(x))\\
       \end{array}\right.
\end{equation}

where $\pi:W\equiv \mathbb{R}\times M^2\to\mathbb{R}\times M$, $(t,x^i,y^j)\mapsto(t,x^i)$.

Let us first introduce an equivalence relation in the set $\mathcal{S}^{\infty}_{ol}({}_\bullet(RF))$ of smooth solutions of ${}_\bullet(RF)$.
\begin{definition}\label{rigid-flow}
Let $(M,\gamma)$ be a Riemannian manifold of dimension $n$. We say that a diffeomorphism $f:M\to M$, is {\em rigid} if $f^*\gamma=\gamma$, and $\det(j(f))=1$, where $j(f)$ is the Jacobian matrix of $f$. In other words $(j(f)^k_i)\in SO(n)$. We say that a flow,  $\phi_t(x)$, solution of ${}_\bullet(RF)$, is {\em rigid}, if $\phi_t^*$, is a rigid diffeomorphism, $\forall t$. Then we say, also, that $\phi_t(x)$ is a {\em rigid solution} of ${}_\bullet(RF)$.
\end{definition}

\begin{definition}\label{equivalence-relation-up-rigid-flow}
We say that two flows $\phi_t(x)$ and $\psi_t(x)$, smooth solutions of ${}_\bullet(RF)$, are {\em rigid equivalent}, if there exists a rigid solution $f_t$ of ${}_\bullet(RF)$ such that $\phi_t=\psi_t\circ f_t$.
\end{definition}

\begin{proposition}\label{equivalence-relation-up-rigid-flow-prposition}
The rigid equivalence is an equivalence relation in $\mathcal{S}^{\infty}_{ol}({}_\bullet(RF))$, that we denote with $\thicksim_R$. Then for any representative $\psi_t\in[\phi_t]_R\in \mathcal{S}^{\infty}_{ol}({}_\bullet(RF))/\thicksim_R\equiv [\mathcal{S}^{\infty}_{ol}({}_\bullet(RF))]_R$, one has
$(\psi_t)^*\gamma=(\phi_t)^*\gamma$, and $\psi_t^*\eta=\phi_t^*\eta$, where $\eta$ is the canonical form associated to the metric $\gamma$.\footnote{The deformation in Fig.\ref{smale-paradox} cannot be rigid since it changes the orientation of the sphere $S^2$. In fact, this deformation can be realized by contracting the Est hemisphere, (resp. West hemisphere), endowed of a normal unitary vector $E$ (resp. $W$), on the middle of its equatorial-line, on the oriented disk $(D^2,E)$, (resp. $(D^2,W)$), with $\partial D^2$ passing for the poles $N$ and $S$. Then, continuing to deform the two oriented disks in the same opposite directions. During this deformation the boundary $\partial D^2$ remains fixed, and both hemispheres conserve their orientations, but the resulting sphere has reversed orientation.}
\end{proposition}

\begin{proof}
In fact, if $\phi_t(x)\in {}_\bullet(RF)$ it follows that  $\phi_t(x)\thicksim_R \phi_t(x)$, since $\phi_t=\phi_t\circ (id_M)_t$. If $\phi_t, \psi_t\in {}_\bullet(RF)$ and $\phi_t\thicksim_R\psi_t$ it follow that there exists a rigid solution $f_t\in {}_\bullet(RF)$ such that $\phi_t=\psi_t\circ f_t$. Since we can also write $\phi_t\circ f^{-1}=\psi_t$, and of course also $f^{-1}\in {}_\bullet(RF)$, it follows that $\psi_t\thicksim_R\phi_t$. Finally, if one has $\phi_t, \psi_t,\varphi_t\in {}_\bullet(RF)$ with $\phi_t\thicksim_R\psi_t\thicksim_R\varphi_t$, it follows that $\phi_t=\psi_t\circ (f_t)_1$, $\psi_t=\varphi_t\circ (f_t)_2$, for some rigid solutions $(f_t)_i$, $i=1,2$. Then we get also
$\phi_t=\varphi_t\circ (f_t)_2\circ (f_t)_1=\varphi_t\circ f_t$, where $f_t=(f_t)_2\circ (f_t)_1$ is also a rigid solution of ${}_\bullet(RF)$. Therefore, $\phi_t\thicksim_R\varphi_t$.
\end{proof}

Let us find solutions of equations(\ref{ricci-flow-equation-b}) of the type

$\phi_t^a=e^{\omega t}h^a(x)$. Then one has that $\omega$ and $h^a(x)$ must satisfy the following equations:

\begin{equation}\label{ricci-flow-equation-c}
          h^a_i(x)h^b_j(x)\left\{\omega\left[2\gamma_{ab}(\phi_t(x))+
          e^{\omega t}h^\alpha(x)(\partial x_\alpha.\gamma_{ab})(\phi_t(x))\right]+2R_{ab}(\phi_t(x))\right\}=0.
\end{equation}

In the case that $(M,\gamma)=(\mathbb{R}^n,\gamma_E)$, then $R_{ab}=0$ and above equations (\ref{ricci-flow-equation-c}) reduce to the following ones

\begin{equation}\label{ricci-flow-equation-d}
          h^a_i(x)h^b_j(x)\omega\left[2\gamma_{ab}(\phi_t(x))+
          e^{\omega t}h^\alpha(x)(\partial x_\alpha.\gamma_{ab})(\phi_t(x))\right]=0.
\end{equation}
Let us add the condition $\phi_0=id_M$, then we get that necessarily $h^a(x)=x^a$, for $a=1,\dots, n$.  Therefore, equations (\ref{ricci-flow-equation-d}) reduce to the following ones:

\begin{equation}\label{ricci-flow-equation-e}
          \omega\left[2\gamma_{ij}(\phi_t(x))+
          e^{\omega t}x^\alpha(\partial x_\alpha.\gamma_{ij})(\phi_t(x))\right]=0.
\end{equation}

Of course we can take a cartesian coordinate system, so that $\gamma_{ij}=\delta_{ij}$. Therefore, equations (\ref{ricci-flow-equation-e}) reduce to the following equations $2\omega\delta_{ij}=0$. Thus we get the unique trivial solution for $\omega$, i.e., $\omega=0$ and the unique solution for $\phi_t(x)$, i.e., $\phi_t^a=x^a$, that gives the unique solution $g(x,t)=\gamma_E(x)$.

Above result does not depend on the particular assumption made on the type of solutions. In fact, if consider equation (\ref{ricci-flow-equation-b}) in the case $M=\mathbb{R}^n$, with $\gamma=\delta_{ij}$, we get the following equations

\begin{equation}\label{ricci-flow-equation-f}
\left\{\begin{array}{l}
A^{rs}_{abij}(\partial t\partial x_r.\phi_t^a)(\partial x_s.\phi_t^b)=0\\
A^{rs}_{abij}\equiv\delta_{ab}(\delta^{rs}_{ij}+\delta^{rs}_{ji}).\\
\end{array}
\right.\end{equation}

Then, one can easily see that the unique smooth solution of (\ref{ricci-flow-equation-f}) that satisfies the initial condition $\phi^a_0=x^a$ is just $\phi_t^a=x^a$, for all $t\in\mathbb{R}$. (Of course this is up to rigid flows that do not produce any deformation.) In fact, let us write above equation for $1 \le i=j\le n$. We get the following equations:

\begin{equation}\label{ricci-flow-equation-g}
\left\{\begin{array}{l}
2\delta_{ab}f^a_{t1}f^b_1=0\\
\cdots\\
2\delta_{ab}f^a_{tn}f^b_n=0\\
\end{array}
\right.\end{equation}

Here, for simplicity, we have put $f\equiv\phi_t$. Then we can also write:
\begin{equation}\label{ricci-flow-equation-h}
\left\{\begin{array}{l}
\partial t.\left((\partial x_1.f^1)^2+\cdots+(\partial x_1.f^n)^2\right)=0\\
\cdots\cr
\partial t.\left((\partial x_n.f^1)^2+\cdots+(\partial x_n.f^n)^2\right)=0\\
\end{array}
\right.\end{equation}

Therefore, we get the following first integrals:
\begin{equation}\label{ricci-flow-equation-h}
\left\{\begin{array}{l}
(\partial x_1.f^1)^2+\cdots+(\partial x_1.f^n)^2=c_1(x^1,\cdots,x^n)\\
\cdots\cr
(\partial x_n.f^1)^2+\cdots+(\partial x_n.f^n)^2=c_n(x^1,\cdots,x^n)\\
\end{array}
\right.\end{equation}

This means that the jacobian matrix $(\partial x_i.(\phi_t)^j)=(j^j_i(x^1,\cdots,x^n))$ is made by functions that depend only on the coordinates $x^i$.
\begin{equation}\label{ricci-flow-equation-i}
(\partial x_i.(\phi_t)^j)=j^j_i(x^1,\cdots,x^n),\quad i,j=1,\cdots, n.
\end{equation}

Let us integrate (\ref{ricci-flow-equation-i}):

\begin{equation}\label{ricci-flow-equation-l}
(\phi_t)^j=\int j^j_i(x^1,\cdots,x^n)\, dx^i+c^j(t,x^1,\cdots,\hat x^i,\cdots,x^n),
\end{equation}
where $c^j(t,x^1,\cdots,\hat x^i,\cdots,x^n)$ is any function that does not depend on $x^i$.
Taking into account the initial condition $\phi^j_0=x^j$, we get
\begin{equation}\label{ricci-flow-equation-m}
x^j=\int j^j_i(x^1,\cdots,x^n)\, dx^i+c^j(0,x^1,\cdots,\hat x^i,\cdots,x^n).
\end{equation}
Let us again derive (\ref{ricci-flow-equation-m}) with respect to $x^i$, we get $\delta^j_i=j^j_i$. So we have
$(\phi_t)^j=x^j+c^j(t,x^1,\cdots,\hat x^i,\cdots,x^n)$. Since this holds for any $1\le i\le n$, we conclude that the arbitrary functions $c^j$ can depend only on $t$. So we have the following functions:
\begin{equation}\label{ricci-flow-equation-n}
(\phi_t)^j=x^j+c^j(t).
\end{equation}
The flows given in (\ref{ricci-flow-equation-n}) are rigid flows, therefore do not produce any deformation. Since, of course, we aim find solutions up to rigid ones, we get that in the case of the $n$-dimensional Euclidean manifold $(\mathbb{R}^n,\gamma)$, the unique complete noncompact solution of Ricci flow equation is just $g(t,x)=\gamma$.
\end{proof}

\section{\bf PINCHING PROBLEMS IN RICCI FLOW EQUATION}

 In the pinching problems connected to the Ricci flow equation one aims to characterize solutions of this equation on a connected compact Riemannian manifold $(M,\gamma)$ with respect to the so-called {\em pinching constant} $\lambda=\hbox{\rm min sec/max sec}$, i.e., by adding an upper as well as a lower bound on the sectional curvature (denoted {\rm sec}). With respect to this scalar-characterization of Riemannian manifolds, there are some interesting informations on their global structure. Let us recall some ones.\footnote{These results are usually referred as {\em sphere theorems}, or {\em quarter-pinched sphere theorems}. In fact, they are generalizations of the {\em sphere theorem for $3$-manifolds} that states that if $M$ is an orientable $3$-manifold with $\pi_2(M)\not=0$, then there exists a non-zero element of $\pi_2(M)$ having a representative that is an {\em embedding} $S^2\to M$.} (S. Brendle and R. Schoens \cite{BRENDLE-SCHOENS}) If $M$ is a compact $n$-dimensional Riemannian manifold with $\lambda\in({{1}\over{4}},1)$ ({\em strict pinching}), $M$ is diffeomorphic to a spherical space form. If $\lambda\in[{{1}\over{4}},1]$ $M$ is diffeomorphic to a spherical space form or isometric to a locally symmetric space.
 (M. Berger \cite{BERGER} and W. Klingenberg \cite{KLING}) If $M$ is a compact simply connected manifold with $\lambda\ge{{1}\over{4}}$ then $M$ is either homeomorphic to $S^n$ or isometric to $\mathbb{C}P^n$, $\mathbb{H}P^n$ or $Ca\mathbb{P}^2$, with their standard Fubini metric. [(J. L. Synge \cite{SYNGE}) A manifold with positive curvature does not necessitate be simply connected. In fact if $\dim M=n=2k$, one has $\pi_1(M)=0$ if orientable and $\pi_1(M)=\mathbb{Z}_2$ if non-orentable. If $\dim M=n=2k+1$ and positively curved $M$ is orientable.] (J. Cheeger \cite{CHEEGER}) Given a constant $\epsilon>0$, there are only finitely many diffeomorphism types of compact simply connected $2n$-dimensional manifolds $M$ with $\lambda\ge\epsilon$. (F. Fang and X. Rong \cite{FANG-RONG}, A. Petrunin and W. Tuschmann \cite{PETRUNIN-TUSCHMANN, TUSCH}) Given a constant $\epsilon>0$, there are only finitely many diffeomorphism types of compact $(2n+1)$-dimensional manifolds  $M$ with $\pi_1(M)=\pi_2(M)=0$ and $\lambda\ge\epsilon$.

Let us consider four-dimensional manifolds with {\em $\lambda$-pinched flag curvature}, ($0\le \lambda<1$), i.e., $R_u(v,v)\ge\lambda(x)R_u(w,w)$, where $R_u(.,.)=R(u,.,u,.)$ is the symmetric bilinear form, identified by the curvature tensor $R$ and any nonzero vector $u\in T_xM$. Here $v,w\in T_x^\perp<\mathbb{R}u>$, $|v|=|w|$. In \cite{ANDREWS-NGUYEN} B. Andrwes and H. Nguyen proved that the class of positively curved compact connected $4$-dimensional manifolds, with $\lambda$-pinched flag curvature, $\lambda\ge {{1}\over{4}}$, is invariant under Ricci-flow. Moreover, any such manifold is either diffeomorphic to a spherical space form or isometric to $\mathbb{C}P^2$ with Fubini-study metric (up to scaling).
In that interesting paper, authors refer to a uniqueness existence theorem for local solutions of the Cauchy problem for the Ricci-flow equation. (A first proof has been given by R. S. Hamilton \cite{HAMIL1, HAMIL2}. Different proofs were obtained also by D. T. De Turk \cite{DETURK} and B. Chow and D. Knopp \cite{CHOW-KNOPP}.) However, it is important to underline that such uniqueness is strictly related to the class of regular solutions considered there. In fact, by recasting the Cauchy problem in the geometric theory of PDE's, one can see that the Ricci-flow equation identifies an analytic submanifold $(RF)\subset J{\it D}^2(W)$ of the second jet-derivative space for sections of the following fiber bundle $\pi:W\equiv \mathbb{R}\times \widetilde{S^0_2(M)}\to \mathbb{R}\times M$, where $\widetilde{S^0_2(M)}\subset S^0_2(M)$ is the open subbundle of non-degenerate symmetric tensors of type $(0,2)$ on $M$. (See \cite{PRA14} to understand in which sense must be interpreted the statement about uniqueness of smooth solutions for the Ricci flow equation.) However, in order to describe singularities in the flow, it is useful to consider the embeddings $(RF)\subset J{\it D}^2(W)\subset J^2_{n+1}(W)$, where $J^2_{n+1}(W)$ is the $2$-jet-space for $(n+1)$-dimensional submanifolds of $W$, $\dim M=n$. Since the Ricci-flow equation is formally integrable and completely integrable, with non-trivial symbol $g_2$, there are also singular solutions satisfying smooth Cauchy problems.\footnote{It is well known that every compact smooth manifold, (or compactifiable $C^\infty$ manifold), can be considered analytic too. This is exentially the meaning of the famous results by J. Nash \cite{NASH} and some relative improvements by A. Tognoli \cite{TOGNOLI} and T. Kawakami \cite{KAWA}.} Then the characterization of global solutions is obtained by means of the singular integral bordism groups of $(RF)$. (For details on the geometry of PDE's see \cite{PRA1, PRA2, PRA3, PRA4, PRA5, PRA6, PRA7, PRA8, PRA9, PRA10, PRA11, PRA14}.) For example, in the particular case of $4$-dimensional closed, compact, simply connected, smooth Riemannian manifolds $M$, one obtains $\Omega_4^{(RF)}/K^{(RF)}_4\cong\mathbb{Z}_2\oplus\mathbb{Z}_2\cong \Omega_4$, where $K^{(RF)}_4$ is the kernel of the canonical projection $p:\Omega_4^{(RF)}\to\Omega_4\cong \mathbb{Z}_2\bigoplus\mathbb{Z}_2$. With this respect, and taking into account the well-known theorems by A. Dold \cite{DOLD} and C. T. C. Wall \cite{WALL} on the (oriented) cobordism ring (${}^{+}\Omega_\bullet$) $\Omega_\bullet$, we can get from the results of this paper the following interesting theorem.

\begin{theorem}\label{}
The $4$-dimensional Riemannian manifolds preserving $\lambda$-pinched flag curvature, $\lambda\ge {{1}\over{4}}$, in a Ricci-flow, belong to the cobordism classes in the image $p^{-1}(r({}^{+}\Omega_4\cong\mathbb{Z}))\subset \Omega_4^{(RF)}$, where $r:{}^{+}\Omega_\bullet\to\Omega_\bullet$ is the forgetting orientation natural mapping, i.e. one has the exact sequences {\em(\ref{exact-commutative-diagram-oriented-non-oriented-bordisms-integral-bordism})}.
\begin{equation}\label{exact-commutative-diagram-oriented-non-oriented-bordisms-integral-bordism}
 \xymatrix{&&&\fbox{${}^+\Omega_4\cong\mathbb{Z}$}\ar[d]^{2}&\\
 &&&\fbox{${}^+\Omega_4\cong\mathbb{Z}$}\ar[d]^{r}&\\
 0\ar[r]&K_4^{(RF)}\ar[r]&\Omega_4^{(RF)}\ar[r]_(0.3){p}&\fbox{$\Omega_4\cong\mathbb{Z}_2\bigoplus\mathbb{Z}_2$}\ar[r]&0}.
\end{equation}
\end{theorem}

\section{\bf NECKPINCHING PROBLEMS IN SINGULAR EXOTIC HEAT PDE}\label{neckpinching-problems-in-singular-exotic-heat-pde}

In a recent paper Z. Gang and I. M. Sigal \cite{GANG-SIGAL} considered some particular solutions for the Ricci flow equation encoding the mean curvature flow of an initial hypersurface $M_0\subset \mathbb{R}^{d+1}$, that is of revolution around the axis $x=x_{d+1}$. There $u(x,t)$ represents, at fixed $t$, the ''distance function'' of the revolution hypersurface from the revolution axis at $x\in\mathbb{R}$. More precisely they considered the following boundary value problem:
\begin{equation}\label{singular-exotic-heat-pde}
\fbox{$\begin{array}{l}
  \hbox{\rm(mean curvature flow equation):}\hskip 2pt u_t={{u_{xx}^2}\over{1+(u_x)^2}}-{{d-1}\over{u}} \\
  \\
\hbox{\rm(boundary conditions):}\, \left\{\begin{array}{l}
u(x,0)=u_0(x)>0,\hskip 2pt \forall x\in\mathbb{R}\\
\lim \inf_{|x|\to\infty} u_0(x)>0\\
||{{1}\over{u(x,t)}}||_\infty<\infty, t<t^*\\
||{{1}\over{u(x,t)}}||_\infty\to\infty, t\to t^*\\
                         \end{array}\right.\\
\end{array}$}
\end{equation}

They proved, by using classic method of functional analysis, existence of solutions that in finite time collapse on the revolution axis. They called such solutions {\it collapsing} (or {\em neckpinching}) at the time $t^*$.

It is interesting to emphasize that translating above problem in the framework of the geometry of PDE's, the mean curvature flow equation in (\ref{singular-exotic-heat-pde}), identifies a real analytic $7$-dimensional submanifold $E_2$ of the $2$-jet-derivative space $J{\it D}^2(W)\cong \mathbb{R}^8$, $(t,x,u,u_t,u_x,u_{tt}, u_{tx},u_{xx})$, over the trivial vector bundle $\pi:W\equiv \mathbb{R}^3\to \mathbb{R}^2$, $(t,x,u)\mapsto (t,x)$. One has $E_2\cap S_2=\emptyset$, where $S_2\subset J{\it D}^2(W)$ is the analytic submanifold identified by the constraint $u=0$. So, a solution $V_\bullet\subset E_2$, such that $u(x,t^*)=0$, for some time $t^*$, cannot exist. However, this does not exclude that by considering the {\it singular equation} ${}^\bullet E_2\equiv E_2\bigcup S_2\subset J{\it D}^2(W) $, we can find {\it asymptotic solutions} of the above problem. This is in fact the case when $S_2$ is endowed with the distribution obtained from the Cartan distribution of $J{\it D}^2(W)$, just restricted on $S_2$. Then one can see that one can find solutions of $E_2$ that approach integral manifolds of $S_2$, when $u\to 0$. The tangent space to such solutions have zero time component in the soldering points. This just means that such solutions stop (or collapse), in a finite time $t^*$, to the revolution axis. (For informations on this geometric approach to singular PDE's see \cite{PRA3, PRA14, PRA15, AG-PRA2}.)

The Cartan distribution $\mathbf{E}_2\subset TE_2$ of $E_2$, is given by the following vector fields
\begin{equation}\label{cartan-distribution-2-a}
\begin{array}{ll}
  \zeta&=X^t(\partial t+u_t\partial u+u_{tt}\partial u_t+u_{tx}\partial u^x)\\
  &+X^x(\partial x+u_x\partial u+u_{xt}\partial u_t+u_{xx}\partial u^x)\\
  &+Z_{xx}\partial u^{xx}+Z_{tt}\partial u^{tt}+Z_{tx}\partial u^{tx}\\
  \end{array}
\end{equation}

such that
\begin{equation}\label{cartan-distribution-2-b}
\left\{
\begin{array}{l}
X^t\left[u_t(u_t(1+u_x^2)-u^2_{xx})+u_{tt}(u(1+u^2_x))+u_{tx}(2uu_tu_x+2u_x(d-1))\right] \\
+ X^x\left[u_x(u_t(1+u_x^2)-u^2_{xx})+uu_{tx}(1+u^2_x)+u_{xx}(2uu_tu_x+2u_x(d-1))\right] \\
+2Z_{xx}uu_{xx}=0\\
\end{array}
\right\}
\end{equation}
with
\begin{equation}\label{cartan-distribution-2-ab}
 u_t=\frac{uu_{xx}^2-(1+u^2_x)(d-1)}{u(1+u^2_x)}.
 \end{equation}
For $u\to 0$ equation (\ref{cartan-distribution-2-b}) becomes $0=X^t[(1+u^2_x)(d-1)]^2$, hence we get $X^t=0$. Therefore, for $u\to 0$ we can write (\ref{cartan-distribution-2-a}) in the form given in (\ref{cartan-distribution-2-aa}).
\begin{equation}\label{cartan-distribution-2-aa}
  \zeta=X^x(\partial x+u_x\partial u+u_{xt}\partial u_t+u_{xx}\partial u^x)+Z_{xx}\partial u^{xx}+Z_{tt}\partial u^{tt}+Z_{tx}\partial u^{tx}.
\end{equation}
The Cartan distribution on $S_2$, (i.e., vector fields (\ref{cartan-distribution-2-a}) satisfying the condition $\zeta.u=0$), is given in (\ref{cartan-distribution-2-abb}).
\begin{equation}\label{cartan-distribution-2-abb}
\left\{\begin{array}{ll}
  \zeta&=X^x\left[-\frac{u_x}{u_t}\partial t+\partial x+(u_{tx}-\frac{u_x}{u_t}u_{tt})\partial u_t+(u_{xx}-\frac{u_x}{u_t}u_{tx})\partial u^x\right]\\
  &+Z_{xx}\partial u^{xx}+Z_{tt}\partial u^{tt}+Z_{tx}\partial u^{tx}.\\
  \end{array}\right.
\end{equation}
Therefore, when $q\in S_2$ approaches equation $E_2$, i.e., when $u_t$ approaches the function given in (\ref{cartan-distribution-2-ab}), one has $\lim_{q\to E_2}\mathbf{E}_2(S_2)_q=<\zeta>$, where $\zeta$ are given by the vector fields in (\ref{cartan-distribution-2-abbb})
\begin{equation}\label{cartan-distribution-2-abbb}
  \zeta=X^x\left[\partial x+u_{tx}\partial u_t+u_{xx}\partial u_x\right]+Z_{xx}\partial u^{xx}+Z_{tt}\partial u^{tt}+Z_{tx}\partial u^{tx}.
\end{equation}
This means that $\lim_{q\to E_2}\mathbf{E}_2(S_2)_q\subset\mathbf{E}_2$. This proves that solutions of $E_2$ can be prolonged to an integral manifold of $S_2$. Since, in the asymptotic limit, both distributions have not time components, then the solutions of $E_2$ approach $S_2$ to a fixed $t^*$ and stop on the revolution axis.

By resuming, we get the following theorem, proved by using geometric methods only.

\begin{theorem}\label{schroedinger-equation-global-attractor}
The singular boundary value problem {\em(\ref{singular-exotic-heat-pde})} admits smooth solutions, i.e., smooth solutions of the regular equation $E_2\subset J{\it D}^2(W)$ that collapse in a finite time approaching the set $S_2\subset J{\it D}^2(W) $ of singular points.
\end{theorem}

\section{\bf EXOTIC HEAT-SCR\"ODINGER EQUATION}

In \cite{MOLINET} it is studied the following so-called {\em cubic nonlinear Schr\"odinger equation} (NLS):
\begin{equation}\label{heat-schroedinger-equation}
\left\{\begin{array}{l}
 u_t+\gamma u+i(u_{xx}\mp |u|^2u)=f\\
u(t,x)\in\mathbb{C},\quad\forall(t,x)\in\Omega\equiv\mathbb{R}_+\times \mathbb{T},\hskip 2pt \mathbb{T}\equiv \mathbb{R}/2\pi\mathbb{Z}\\
\end{array}\right.
\end{equation}

where $\gamma>0$ ({\em damping parameter}) and $f\in L^2(\mathbb{T})$ ({\em time-independent-forcing term}). There the main result is that the nonlinear group $S(\cdot)$, associated to (\ref{heat-schroedinger-equation}), i.e., $S(t)(u_0)=u(t)$, $t\in\mathbb{R}$, where $u$ is the solution of (\ref{heat-schroedinger-equation}), for to the initial condition $u_0$, provides an infinite-dimensional dynamic system in $L^2(\mathbb{T})$ that has a global attractor $\mathcal{A}\subset H^2(\mathbb{T})$, that is a connected and compact set of $H^2(\mathbb{T})$, invariant (positively and negatively) by $S(\cdot)$ that attracts for the $L^2(\mathbb{T})$-metric all positive orbits uniformly with respect to bounded sets of initial data in $L^2(\mathbb{T})$.

In the framework of the algebraic topology of PDE's,  the interesting problem considered in that paper, has an intriguing issue. With this respect, let us recast equation (\ref{heat-schroedinger-equation}) into a real analytic PDE of second order over the following trivial vector fiber bundle $\pi:W\equiv M\times \mathbb{R}^2\to M\equiv \mathbb{R}_+\times\mathbb{T}$, $(t,x,v,w)\mapsto(t,x)$:
\begin{equation}\label{real-heat-schroedinger-equation}
  \fbox{$E_2\subset J\mathcal{D}^2(W)\hskip 2pt:\left\{
\begin{array}{l}
 v_t+\gamma v-w_{xx}\mp w(v^2+w^2)=\phi\\
 w_t+\gamma w+v_{xx}\mp v(v^2+w^2)=\psi\\
\end{array}
\right.$}
\end{equation}

where $u=v+iw$ and $f=\phi+i\psi$. Here we assume that $\phi$ and $\psi$ are analytic functions. Then one can prove that $E_2$ is an involutive, formally integrable and completely integrable PDE. In fact, one has
\begin{equation}\label{formal-integrability-real-heat-schroedinger-equation}
\fbox{$\dim(E_2)_{+1}=16$}=\fbox{$\dim E_2=12$}+\fbox{$\dim(g_2)_{+1}=4$}.
\end{equation}
The relation (\ref{formal-integrability-real-heat-schroedinger-equation}) proves that the canonical mapping $(E_2)_{+1}\to E_2$ is surjective. Furthermore, since $\dim(g_2)_{+1}=4=\dim g_2$, we get that $g_2$ is an involutive symbol. This facts are enough to state that $E_2$ is formally integrable, and since it is analytic it is completely integrable too. This means that in the neighborhood of any point $q\in E_2$, ({\em initial condition}), passes a regular solution (analytic solution). Furthermore, since $E_2$ satisfies some conditions of regularity, we can solve Cauchy problems for $1$-dimensional integral manifolds $N\subset E_2$, diffeomorphically projected in $W$, by means of the canonical projection $\pi_{2,0}:E_2\to W$. We call {\em admissible} such Cauchy manifolds. Note that solutions passing through admissible Cauchy manifolds do not necessitate to be smooth, in fact, in general, are singular ones with respect to the embeddings $E_2\subset J{\it D}^2(W)\subset J_2^2(W)$, where $J_2^2(W)$ denotes the $2$-jet-space for $2$-dimensional  submanifolds of $W$.

Furthermore, one can see that the $1$-dimensional integral singular bordism group of $E_2\subset J^2_2(W)$ is $\Omega_{1,s}^{E_2}\cong\mathbb{Z}_2$. Then for any two space-like smooth (or analytic) $1$-dimensional closed admissible Cauchy hypersurfaces $N_1\subset (E_2)_{t_1}$ and $N_2\subset (E_2)_{t_2}$, $t_1\not= t_2$, where $(E_2)_t\equiv \bar\pi_2^{-1}(t)$, with $\bar\pi$ the natural projection $E_2\to \mathbb{R}_+$, exists a singular solution $V\subset E_2$, such that $\partial V=N_1\sqcup N_2$, iff $[N_1\sqcup N_2]=[0]\in\Omega_{1,s}^{E_2}$. In order that this condition should be satisfied, it is enough that $N_1$ and $N_2$ have the same integral characteristic numbers of second order. (The solution $V$ bording $N_1$ and $N_2$ is a smooth solution iff above condition holds for all the orders, i.e. for all the conservation laws of $E_2$.) So, in general, there is not solution unicity for any admissible, $1$-dimensional closed smooth space-like Cauchy manifold, $N\subset E_2$, but all such solutions bord ones belonging to the same $0$-class in $\Omega_{1,s}^{E_2}\cong\mathbb{Z}_2$. This agrees with the main result in \cite{MOLINET}. By resuming we get the following theorem.\footnote{Here we have explicitly considered only the analytic case. However, it is also possible to extend such result to singular PDE's, similarly to what made in the previous Section \ref{neckpinching-problems-in-singular-exotic-heat-pde}.}

\begin{theorem}
Equation {\em(\ref{real-heat-schroedinger-equation})}, considered in the analytic case, has a {\em global attractor}, in the sense that all its (singular) solutions bord with a same integral bordism class.
\end{theorem}

\section{\bf EXOTIC SINGULAR VECTOR HEAT EQUATION}

In \cite{DASKA-SESUM} it is studied the PDE reported in (\ref{exotic-vector-heat-equation}).

\begin{equation}\label{exotic-vector-heat-equation}
(\partial t.F^k)+\kappa\, \nu^k=0
\end{equation}

where $F^k=F^k(t,u^a)\equiv F^k_t$, is a family of parametric equations for embeddings $F_t:\Sigma^2\to\mathbb{R}^3$ of $2$-dimensional, convex, star-shaped, compact surface $\Sigma^2$ into $\mathbb{R}^3$, with $(u^a)_{1\le a\le 2}$ local coordinates on $\Sigma^2$ and $(x^k)_{1\le k\le 3}$, coordinates in $\mathbb{R}^3$. $\kappa=\kappa(t,u^a)\equiv \hbox{\rm Gauss curvature}/\hbox{\rm mean curvature}$ denotes the so-called {\em harmonic mean curvature} of $\Sigma_t\equiv F_t(\Sigma^2)\subset\mathbb{R}^3$. $\nu=\nu^k\partial x_k$ is the unitary normal vector field on $\Sigma_t$. In \cite{DASKA-SESUM} it is assumed the mean curvature positive. The main result there concerns an existence of solutions $\Sigma_t^\epsilon$ for short time starting from a $\Sigma^2$ of class $C^{2,1}$ and a maximal time $T_\epsilon$ of existence of a smooth solution such that the mean curvature goes to zero as $t\to T_\epsilon$ at some point of $\Sigma_t^\epsilon$, or $\Sigma_t^\epsilon$ shrinks to a point as $t\to T_\epsilon$. In the particular case where $\Sigma^2$ is a surface of revolution, the flow always exists up to the time when the surface shrinks to a point.
It is interesting to recast this problem in the framework for geometric theory of PDE's, i.e., to implement this problem on a trivial vector fiber-bundle $\pi:W\equiv \mathbb{R}^6\to\mathbb{R}^3$, $(t,u^a,x^k)\mapsto(t,u^a)$. Then the problem is encoded by the PDE reported in (\ref{exotic-vector-heat-equation-b}).
\begin{equation}\label{exotic-vector-heat-equation-b}
\fbox{$  E_3\subset J{\it D}^3(W):\hskip 2pt\left\{
\begin{array}{l}
 A(x^i,x^i_a,x^i_{ab})x^k_t+B^k(x^i,x^i_a,x^i_{ab})=0\hskip 3pt\hbox{\rm(HMCF)}\\
b_{ab;c}-b_{ac;b}=0\hskip 3pt\hbox{\rm(Gauss-Codazzi equation)}\\
b_{ab}=[\epsilon_{ijk}x^i_{ab}x^j_{\bar a}x^k_{\bar b}\delta^{\bar a}_1\delta^{\bar b}_2]/\sqrt{EG-F^2}.\\
\end{array}
\right.$}
\end{equation}

In (\ref{exotic-vector-heat-equation-b}) $A$ and $B^k$ are known analytic functions of their arguments, $b_{ab}$ are the components of the second fundamental form and $E,F,G$, are the usual Gauss symbols of the surface $\Sigma_t$. The semi-colon, in the Gauss-Codazzi equation denotes tensor derivative with respect to $u^c$.  In general the PDE in (\ref{exotic-vector-heat-equation-b}) is a singular equation, however in the case considered in \cite{DASKA-SESUM}, i.e., positive mean curvature, one has $A\not=0$. Then, in order to characterize local solutions, it is important to study the formal properties of such equation, and, by means the determination of its integral bordism groups, characterize global solutions also. In this way one could generalize the following well known result of differential geometry: {\em Given symmetric functions $\gamma_{ab}=\gamma_{ab}(u^c)$ and $b_{ab}=b_{ab}(u^c)$, $a,b,c=1,2$, such that the Gauss-Codazzi equations are satisfied, there exists a surface $x^i=x^i(u^c)$, uniquely determined up to rigid motions, such that $\gamma$ and $b$ are respectively the first fundamental form and the second fundamental form of such a surface.} (See, e.g. L. P. Eisenhart \cite{EISEN}.) Of course this agrees with the structure of equation in (\ref{exotic-vector-heat-equation-b}), since there coordinates with time derivatives are determined by the other ones containing coordinates with space derivatives only. It is important to emphasize that equation (\ref{exotic-vector-heat-equation-b}) is not formally integrable, since there is not surjectivity between the first prolongation $(E_3)_{+1}$ and $E_3$. In fact,
\begin{equation}\label{non-formal-integrability-vector-singular-pde}
\fbox{$\dim((E_3)_{+1})=88$}<\fbox{$\dim (E_3)=58$}+\fbox{$\dim((g_3)_{+1})=41$}.
\end{equation}

However, by using the geometric theory of PDE's, it is possible to identify a subequation $\widehat{(E_3)}\subset E_3$ that is formally integrable and completely integrable. For this it is enough to add to the equations in (\ref{exotic-vector-heat-equation-b}) also the first prolongations of (HMCF). More precisely, we shall use equation (\ref{exotic-vector-heat-equation-c}).

\begin{equation}\label{exotic-vector-heat-equation-c}
\fbox{$  \widehat{E_3}\subset J{\it D}^3(W):\hskip 2pt\left\{
\begin{array}{l}
 Ax^k_t+B^k=0\\
 A x^k_{tt}+[(\partial x^a_i.A)x^i_{at}+(\partial x^{ab}_i.A)x^i_{abt}]x^k_t\\
 \hskip 5pt+(\partial x^a_i.B^k)x^i_{at}+(\partial x^{ab}_i.B^k)x^i_{abt}=0\\
  A x^k_{tc}+[(\partial x^a_i.A)x^i_{ac}+(\partial x^{ab}_i.A)x^i_{abc}]x^k_t\\
   \hskip 5pt+(\partial x^a_i.B^k)x^i_{ac}+(\partial x^{ab}_i.B^k)x^i_{abc}=0\\
b_{ab;c}-b_{ac;b}=0\\
{[b_{ab;c}-b_{ac;b}]_{,t}=0}\\
{[b_{ab;c}-b_{ac;b}]_{,d}=0}\\
{b_{ab}=[\epsilon_{ijk}x^i_{ab}x^j_{\bar a}x^k_{\bar b}\delta^{\bar a}_1\delta^{\bar b}_2]/\sqrt{EG-F^2}}.\\
\end{array}
\right.$}
\end{equation}
Then $\widehat{E_3}_{+1}\to\widehat{E_3}$ is surjective as proved in (\ref{formal-integrability-vector-singular-pde}).
\begin{equation}\label{formal-integrability-vector-singular-pde}
\fbox{$\dim((\widehat{E_3})_{+1})=70$}=\fbox{$\dim \widehat{E_3}=49$}+\fbox{$\dim((\widehat{g_3})_{+1})=21$}.
\end{equation}

Furthermore, the symbol $\widehat{g_3}$ is involutive, since one has $\dim(\widehat{g_3}_{+1})=21=\dim(\widehat{g_3})$. Therefore $\widehat{E_3}$ is formally integrable and completely integrable.

Then the characterization of Cauchy problems and global solutions, weak, singular and smooth, can be directly obtained by using the algebraic topologic methods in PDE's  introduced by A. Pr\'astaro. The asymptotic behaviour of solutions as mean curvature approaches the zero value, can be obtained by considering the singular PDE in (\ref{singular-vector-singular-pde})
\begin{equation}\label{singular-vector-singular-pde}
\fbox{$\widehat{(E_3)}^{(S)}\equiv \widehat{(E_3)}\bigcup S_3\subset J{\it D}^3(W):\hskip 2pt\left\{\begin{array}{l}
                                                                      \widehat{(E_3)} \hskip 0.5cm\hbox{\rm(\ref{exotic-vector-heat-equation-c})}  \\                 S_3\equiv\{{{1}\over{2}}\gamma^{ab}b_{ab}=0\}\\
                                                                      \end{array}\right.$}
\end{equation}

i.e., adding to equation (\ref{exotic-vector-heat-equation-c}) the submanifold $S_3$ of $J{\it D}^3(W)$, identified by the condition of zero mean-curvature. (See, e.g. Section \ref{neckpinching-problems-in-singular-exotic-heat-pde} where similar problems are considered.)

\section{\bf COMPLEX RICCI FLOW EQUATION AS A QUANTUM EQUATION AND POINCAR\'E CONJECTURE}

In (\ref{complex-ricci-flow-equation}) it is reported the so-called {\em normalized K\"ahler-Ricci flow equation}
\begin{equation}\label{complex-ricci-flow-equation}
\scalebox{0.9}{$ \left\{\begin{array}{l}
\hbox{\rm(normalized K\"ahler-Ricci flow equation)}:\hskip 2pt \partial_t.g(t)=-Ric(g(t))+g(t)\\
\hbox{\rm $g(t)$ is a K\"ahler metric for each $t$;}\\
\hbox{\rm $\omega(t)=\omega_{g(t)}$, the K\"ahler form associated to $g(t)$, satisfies the condition}\\
\hskip 5pt[\omega(t)={{i}\over{2\pi}}g(t)_{\alpha\bar\beta}dz^\alpha\wedge d\bar z\beta ]=c_1(M)
=[{{i}\over{2\pi}}R_{\alpha\bar\beta}(g(t))dz^\alpha\wedge d\bar z\beta =\rho(t)]>0.\\
\end{array}\right.$}
\end{equation}
Here $M$ is a compact K\"ahler manifold, with the first Chern  class $c_1(M)>0$. (Positivity of the first Chern class is in the sense of K. Kodaira \cite{KODAIRA}.) $\omega(t)$ denotes the K\"ahler form associated to $g(t)$ and $\rho(t)$ is the Ricci form that represents also the first Chern class of $M$. Some authors study solutions of (\ref{complex-ricci-flow-equation}) by adding also the global constraint in (\ref{bounded-problem}).
\begin{equation}\label{bounded-problem}
   \int_M|Rm(g(t))|^{n/2}\, dv_t\le C,\hskip 2pt \dim_{\mathbb{C}}M=n\ge 3,\hskip 2pt C\in\mathbb{R}
\end{equation}

There $Rm(g(t))$ denotes the curvature operator. (See, e.g., the work by P. Daskalopoulos and M. Sesum \cite{DASKA-SESUM}, and papers quoted there.) The main result in \cite{DASKA-SESUM} is an improvement of a previous result by Sesum, where was also made the hypothesis of bounded Ricci curvature: $|R(g(t)|\le C$. More precisely if $g(t)$, $t\in[0,\infty)$ is a solution of the problem (\ref{bounded-problem}) that at $t=0$ has curvature operator uniformly bounded in $L^n$-norm, the curvature operator will also be uniformly bounded along the flow. This is enough to state that the flow will converge along a subsequence to a K\"ahler-Ricci soliton, i.e., a compact K\"ahler manifold $(N,h)$, such that its K\"ahler form $\omega_h$ satisfies the following equation: $Ric(\omega_h)-\omega_h=\mathcal{L}_\zeta\omega_h$, where $\zeta$ is a holomorphic vector field on $N$, and $\mathcal{L}_\zeta$ denotes Lie derivative with respect to $\zeta$. It follows that the first Chern class $c_1(N)$ of $N$ is positive and represented by $\omega_h$. Since there are no K\"ahler-Einstein metrics on $N$ if $N$ admits a K\"ahler-Ricci soliton, (this is a result by A. Futaki \cite{FUTAKI}), it follows that the existence of K\"ahler-Ricci soliton is an obstruction to the existence of K\"ahler-Einstein metrics on compact K\"ahler manifolds with positive first Chern class. The condition $c_1(M)>0$ is important, after the result by S.-T. Yau \cite{YAU1}, on the existence of K\"ahler-Einstein metrics on K\"ahler manifolds with $c_1(M)\le 0$, and by T. Aubin \cite{AUBIN} for K\"ahler manifolds with $c_1(M)<0$. (This was first conjectured by E. Calabi \cite{CALABI}. For complementary informations see, e.g., \cite{YAU2}.)

It is interesting to emphasize that the problem considered in (\cite{DASKA-SESUM}) can be recast in the geometric theory of PDE, directly working in the category of complex manifolds. This can be made, by using the geometric theory of quantum PDE's formulated by A. Pr\'astaro. In fact the category of complex manifolds can be considered a little subcategory of the one for quantum manifolds, (in the sense introduced by A. Pr\'astaro), where the quantum algebra coincides with the algebra of complex numbers and quantum differentiable functions are identified with holomorphic functions. Then the system (\ref{complex-ricci-flow-equation}) can be encoded with the following second order PDE $\widetilde{E}_2$ in the category of complex manifolds (like quantum manifolds):
\begin{equation}\label{complex-ricci-flow-equation-b}
\scalebox{0.8}{$\fbox{$\widetilde{E}_2\subset J\widetilde{\mathcal{D}}^2(\widetilde{W})\subset \widetilde{J}_{n+1}^2(\widetilde{W}):\left\{
\begin{array}{ll}
 \hbox{\rm(1(a))}&:\hskip 2pt (\partial_t.g)_{\alpha\bar\beta}(t)=-Ric(g(t))_{\alpha\bar\beta}+g(t)_{\alpha\bar\beta}\\
\hbox{\rm(1(b))}&:\hskip 2pt R_{\alpha\bar\beta}=-(\partial z_\alpha\partial\bar z_\beta.\ln\det(g_{\delta\bar\epsilon}))\\
\hbox{\rm (2(a))}&:\hskip 2pt  g_{\alpha\bar\beta}=g_{\bar\beta\alpha}\\
\hbox{\rm (2(b))}&:\hskip 2pt  \bar g_{\alpha\bar\beta}=g_{\bar\alpha\beta}\\
\hbox{\rm (3(a))}&:\hskip 2pt (\partial z_\gamma.g_{\alpha\bar\beta})-(\partial z_\alpha.g_{\gamma\bar\beta})=0\\
\hbox{\rm (3(b))}&:\hskip 2pt  (\partial \bar z_\gamma.g_{\alpha\bar\beta})-(\partial \bar z_\beta.g_{\alpha\bar\gamma})=0\\
\hbox{\rm (4)}&:\hskip 2pt g_{\alpha\bar\beta}-R_{\alpha\bar\beta}-(\partial z_\alpha\partial\bar z_\beta.f)=0\\
\end{array}\right.$}$}
\end{equation}
In (\ref{complex-ricci-flow-equation-b}) $\widetilde{W}$ is the following fiber bundle $\widetilde{\pi}:\widetilde{W}\equiv\mathbb{R}\times\widetilde{S^0_2M}\times\mathbb{C}\to \mathbb{R}\times M$, $(t,z^\alpha,\bar z^\alpha, u_{\alpha\bar\beta},\lambda)\mapsto(t,z^\alpha, \bar z^\alpha)$. (Local) sections of $\widetilde{\pi}$ are identified with the following (local) functions $\{g_{\alpha\bar\beta}(t,z^\alpha,\bar z^\alpha),f(t,z^\alpha,\bar z^\alpha)\}$. The ''tilde'' over the jet(-derivative) spaces denotes ''holomorphic class of differentiability''. Our geometric theory of quantum PDE's allows us to obtain solutions as $(n+1)$-dimensional complex integral submanifolds of $\widetilde{E}_2$. Then the characterization of global solutions is made by means of the integral bordism group $\Omega_{n}^{\widetilde{E}_2}$. With this respect one can generalize the Poincar\'e conjecture in this category of complex manifolds. (For details see \cite{PRA12} where a generalized version of the Poincar\'e conjecture is formulated, and proved too, in the category of quantum (super)manifolds.)

config: arxiv


TITLE: Characteristics for 2nd order differential equations
QUESTION [3 upvotes]: If I have an equation
$p(x)\frac{\partial^2u}{\partial x^2} + r(x)\frac{\partial^2u}{\partial x\partial y} + q(x)\frac{\partial^2 u}{\partial y^2}=f(x,y,u)$
Where $f$ maybe contains first partial derivatives for $u$.
Can anyone give me a worked example of how to solve this using the method of characteristics.
Thanks in advance.

REPLY [2 votes]: See these lecture notes on the method of characteristics for second-order PDEs, where the method is applied to steady isentropic flow (gasdynamics)


TITLE: About extended of continuous function
QUESTION [0 upvotes]: Assume we have an open interval $I\subset\Bbb R$ and a set $D\subset I.$ Let $f\colon D\to \Bbb R$ be a continuous such that $f$ can not be extended to a continuous on $I.$ What we can conclude from that ?

REPLY [1 votes]: $\mathbf{ Claim:}$ The set $ D $ is not close in $ I $.
To see this we proceed by contradiction, in particular we will show that if $ D $ is close then there exists an extension. Suppose $ D $ close and non-empty. Let $ a = \min\{ x \in D \} $ and $ d = \max \{ x \in D \} $. We extend $ f $ in $ \mathbb{R} \setminus [a,b]  $ by setting $ \tilde{f}(x) = f(a) $ for $ x < a $ and  $ \tilde{f}(x) = f(b) $ for $ x > b $. Let $ x \in (a,b) $, we define $ y_-(x) = \sup\{ y \in D $ such that $ y < x \} $ and $ y_+(x) = \inf\{ y \in D $ such that $ y > x \} $. Define
$$ \tilde{f}(x) = f(y_{-}(x)) + \frac{x-y_{-}(x)}{y_{+}(x)-y_{-}(x)}(f(y_{+}(x))-f(y_{-}(x))).$$
The function $ \tilde{f} $ is a continuous extension of $ f $.
$ \textbf{Consequences:} $ if $ f$ cannot be extended to $ I $ then $ f $ cannot be extended to the closure of $ D $ that we denoted $ \bar{D} $.
One can wonder why one cannot extend $ f $ to $ \bar{D} $ and this happens if and only if there exists $ x \in \bar{D} \setminus D $ and a sequence $ x_{n} $ that converge to $ x $ such that $ f(x_{n}) $ does not converge.
$ \textbf{ Some examples:} $

$ f(x_{n}) $ diverge to $ \infty $ (imagine 1/x around zero)

$ f(x_{n}) $ does not converge (imagine a function that oscillate a lot like $sin(1/x)$ close to zero)

$ f(x_{n}) $ converge from the ''left and the right''  but the limit is different (imagine sing(x) around zero)


TITLE: Extension of a radical equals radical of the extension
QUESTION [0 upvotes]: Let $R$ be a commutative ring with identity and let $f\colon R \to R[X]$ be the natural ring homomorphism. Let $I$ be an ideal of $R$. What I have shown is that $I$ is a prime ideal iff $I^e$ is a prime ideal. Now how can I prove
$\sqrt{I^e}=\sqrt{I}^e$? Thanks in advance for your answer.

REPLY [2 votes]: Fact 1: Consider the ring epimomoephis $\psi:R[x]\rightarrow (R/I)[x]$ by $\psi(a_0+a_1x+...+a_nx^n)=(a_0+I)+(a_1+I)x+...+(a_n+I)x^n$. Clearly, $ker(\psi)=I^e=I[x]$. Thus, $R[x]/I[x]\cong (R/I)[x]$.
Fact 2: Let $S$ be a ring. Then $S$ is an integral domain if and only if $S[x]$ is.
Form 1 and 2, we have:
$I$ is a prime ideal of $R$ if and only if $R/I $ is an integral domain if and only if $(R/I)[x]$ is an integral domain if and only if $R[x]/I[x]$ is an integral domain if and only if $I^e=I[x]$ is a prime ideal of $R[x]$.
For the last statment use the fact that $Nil(S[x])=Nil(S)[x]=Nil(S)^e$, where $Nil(A)$ is the set of all nilpotent element of a ring $A$( nilradical of $A$). Thus we have
$f\in \sqrt{I^e}$ if and only if $f^n\in I^e$ for some $n\in \mathbb{N}$ if and only if $f+I^e\in Nil(R[x]/I^e)$ if and only if $\psi(f)\in Nil(R[x]/I[x])$  if and only if $\psi(f)\in Nil(R/I)[x]$ if and only if ( the coefficients of $f$ are in $Nil(R/I)$) if and only if ( the coefficients of $f$ are in $\sqrt{I}$) if and only if $f\in \sqrt{I}^e.$


TITLE: What can be said about strong (p,q) and weak (p,q) properties of Hardy-Littlewood Maximal Function?
QUESTION [2 upvotes]: The question is for 1 $\leq$ p,q $\leq$ $\infty$
I can prove that HL Maximal function is weak (1,1) using the method of finite collection of bounded balls.
I can also prove that it is strong (p,p) as it is a standard proof. However, what can be said about other cases?
Terence Tao's lecture notes make this comment without proof:

Dimensional analysis (analysing how $f$ and $Mf$ react under dilation of the domain $\mathbb R^d$ by a scaling parameter λ) shows that no weak or strong type (p, q) estimates are available in the off-diagonal case p $\neq$ q.

Could this be elaborated upon, perhaps in a mathematically rigorous way? Exactly how to do this dimensional analysis and how does it prove there are no estimates for off-diagonal case?

REPLY [1 votes]: The definition is $$Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}|f(t)|dt$$ where $B(x,r)=\{y \in \Bbb R^n: ||x-y||<r\}$
Then dilate your $f$ by some $\delta>0$, $f^{\delta}(x):=f(\delta x)$

Let's assume that the Hardy-Littlewood Maximal function is strong $(p,q)$. We'll basically show that $p=q$

Now take $L^p$ norms, $||f^{\delta}||_p=(\int_{\Bbb R^n}|f^{\delta}(t)|^pdt)^{1/p}=(\int_{\Bbb R^n}|f(\delta t)|^pdt)^{1/p}$
Now a change of variable $\delta t=y$ gives, $||f^{\delta}||_p=(\int_{\Bbb R^n}|f(\delta t)|^pdt)^{1/p}=({\delta}^{-n}\int_{\Bbb R^n}|f(y)|^pdy)^{1/p}={\delta}^{-n/p}||f||_p \dots (*)$
Now $$Mf^{\delta}(x)=\sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}|f^{\delta}(t)|dt=\sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}|f(\delta t)|dt$$ $$=\sup_{r>0} \frac{{\delta}^{-n}}{|B(x,r)|}\int_{B(\delta x,\delta r)}|f(y)|dy \text{  ( same substitution as above)}$$ $$=\sup_{r>0} \frac{1}{|B(\delta x,\delta r)|}\int_{B(\delta x,\delta r)}|f(y)|dy$$ $$=\sup_{\lambda>0} \frac{1}{|B(\delta x,\lambda)|}\int_{B(\delta x,\lambda)}|f(y)|dy$$ $$=Mf(\delta x)=(Mf)^{\delta}(x)$$
Then $||Mf^{\delta}||_q=(\int_{\Bbb R^n} |Mf^{\delta}(x)|^q dx)^{1/q}= (\int_{\Bbb R^n} |Mf(\delta x)|^q dx)^{1/q}= {\delta}^{-n/q}||Mf||_q \le C \delta^{-n/p}||f||_p$
But then, ${\delta}^{-n(\frac{1}{q}-\frac{1}{p})}=1 \implies p=q$
Now for the weak off-diagonal case, it's a simple application of the Off diagonal version of Marcinkewicz interpolation combining with the fact that the Maximal function is not strong in the off-diagonal setup, which we just proved!


TITLE: Who wins this two player game of making squares?
QUESTION [4 upvotes]: Two players take turns coloring edges on an $n$-by-$n$ grid.  Both players use the same color.  Every time a player surrounds a square of the grid, they mark that square with their name and go again.  A player cannot pass; they must move if it is their turn.  The goal is to make as many squares as possible.  What is the score with optimal play as a function of $n$?  Note that ties are only possible if $n$ is odd so that the total number of squares is even.
To state it more formally: there is a graph with vertices labelled by pairs $(i,j)$ for $i,j \in \{1,\ldots,n\}$.  There is an edge between a pair of vertices $(i,j)$ and $(k,l)\ $ iff $\ |i-k|+|j-l|=1.\ $ The state of the game is described by specifying the player whose move it is and specifying a bit for each edge, saying whether that edge is "colored" or "uncolored".  One player moves first.  Initially all edges are uncolored.  On a turn, a player picks an uncolored edge and colors it.  If the edge that a player colors makes at least one elementary square colored (i.e. that edge is in a cycle of length $4$ of colored edges), then the player scores one point for each such elementary square, and it is that player's turn again.  Otherwise, it is the other players turn. Game terminates when there are no uncolored edges. Each player tries to maximize their own score.

REPLY [13 votes]: This just the game of Dots and Boxes. There is a huge literature on this game. In particular, Berlekamp's book referenced in the above link shows how difficult this game is.


TITLE: Region of integration for double integral:
QUESTION [0 upvotes]: When tasked with reversing the order of integration in the following integral:
$$\int^1_0\int^{y^2}_{-y}x\ \mathrm{d}x\mathrm{d}y$$
It seems sensible to sketch out our situation:
we have that $-y\le x\le y^2$ and that $0 \le y \le 1$ which gives the following enclosed region:

However, where is the fallacy in writing:
$$\int^1_0\int^{y^2}_{-y}x\ \mathrm{d}x\mathrm{d}y = \int^1_0\left[\int^{0.1}_{-y}x\ \mathrm{d}x + \int^{y^2}_{0.1}x\ \mathrm{d}x\right]\mathrm{d}y$$
(Where 0.1 is chosen arbitrarily) which will then give an enlarged region and thus a larger answer:

This seems clearly wrong, but there's something I'm missing when we have both upper and lower limits of an "inner" integral being a function of $y$. If anyone can spot the exact assumption I'm making which is wrong, that would be appreciated.

REPLY [1 votes]: The outer integral is evaluated for $y \in [0,1]$, and the inner integral's limits are on $x \in [-y, y^2]$, which you partition into $x \in [-y, 0.1] \cup (0.1, y^2]$.  So if $y \in [0, \sqrt{0.1})$, the integral over the second interval is negative, because the upper limit of integration is smaller than the lower limit of integration.  For example, if $y = 0.2$, then $y^2 = 0.04 < 0.1$, and the two inner integrals in your expression become $$\begin{align}
\int_{x=-y}^{0.1} x \, dx + \int_{x=0.1}^{y^2} x \, dx &= \int_{x=-0.2}^{0.1} x \, dx + \int_{x=0.1}^{0.04} x \, dx \\
&= \int_{x=-0.2}^{0.1} x \, dx - \int_{x=0.04}^{0.1} x \, dx \\
&= \int_{x=-0.2}^{0.04} x \, dx.
\end{align}$$
This resolves your contradiction.