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                        Problems for thinking

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 Prove that the powers of 3/2 are equally distributed mod 1!

 Prove that
$\(\frac{3}{4}\)^n\le\left\{\(\frac{3}{2}\)^n\right\}\le1-\(\frac{3}{4}\)^n$!
Waring problem for $g\(n\)$ follows!


 Let $\mu_n\(q\)=\frac{1}{n+1}\sum_{k=0}^n \delta_{\left\{q^kright\}}$ be a
sequence of measures for a fixed rational number $q\in {\bf Q}$. What kind
of measures can be a weak${}^\ast$ limits of such sequences? (O'.T)


 Let $F_d(x)=x/d$ if $x=0\quad ({\rm mod}\,\,d)$, $F_d(x)=(d+1)x-j$ if
$x=j\quad ({\rm mod}\,\,d)$ and $F_d(x)=(d+1)x+1$ if 
$x=-1\quad ({\rm mod}\,\,d)$.

 All iterates of $F_d$ for $n\in {\bf Z}_+$ (for any fixed $d\ge 2$)
reach some number smaller than $d$. (Wiggin) 

				     ---

 $T(x)=(m_i x -r_i)/d$, if $x=i\quad ({\rm mod}\,\,d)$, while
$im_i=r_i\quad ({\rm mod}\,\,d)$. This map naturally extends to 
$d$-adic integers. When will $T$ be ergodic?

				     ---

 $a^x+b^y=c^z$, $a$, $b$, $c>0$, $x$, $y$, $z\ge3$ all integers, implies
that $(a,b,c)=1$. (Beal's Conjecture)

				     ---

 Does Penrose tiling 3 colorable? Does all the possible tilings with the
Penrose tiles 3 colorable? (O'.T.)

				     ---

 Prove that the numbers of the form $\sum_{i=1}^n \sqrt{a_i}$, where
$a_i\le2^n$ are positive integers are equal, or have distance at least
$2^{-(1+n)^k)}$ for some integer $k$.

				     ---


 S. Smale, Dynamics retrospective: great problems, attempts that failed.
Nonlinear science: the next decade (Los Alamos, NM, 1990). Phys.-D
[Physica-D.-Nonlinear-Phenomena] 51 (1991), no. 1-3, 267--273.
(From MR:)

 In this paper the author states the following ten problems of dynamical
systems. A short discussion of each is given with some references.

 (1) Is the dynamics of the differential equations in ${\bf R}\sp 3$ of
Lorenz ($\dot x=-10x+10y$, $\dot y=28x-y-xz$, $\dot z=-(\frac83)z+xy)$
described by the ``geometric Lorenz attractor'' of Williams, Guckenheimer
and Yorke?

 (2) Can the Navier-Stokes equations on the two-dimensional torus be
dynamically nontrivial?

 (3) Let $T\colon M\to M$ be Anosov (i.e. globally hyperbolic). Is $T$
topologically the same as the Lie group model of J. Franks?

 (4) Is Axiom A a generic property for one-dimensional dynamical systems?

 (5) Is it a generic property of $T\in{\rm Diff}(M)$ $(C\sp r,\;r>1)$ that
the periodic points are dense in the nonwandering points?

 (6) Is it a generic property that the centralizer $Z(T)$ of 
$T\in{\rm Diff}(M)$ consists of only the iterates of $T$?

 (7) Let $P(x,y)$, $Q(x,y)$ be real polynomials in two variables and
consider the differential equation on ${\bf R}\sp 2$: $dx/dt=P(x,y)$,
$dy/dt=Q(x,y)$. Is there a bound $K$ on the number of limit cycles of the
form $K\leq n^q$, where $n$ is the maximum of the degrees of $P$ and $Q$?

 (8) Given any set of masses, $m\sb 1,\cdots,m\sb n>0$ in the $n$-body
problem of celestial mechanics, is the number of relative equilibria finite?

 (9) Extend the mathematical model of general equilibrium theory to include
price adjustments.

 (10) Extend the dynamics of quantum mechanics in a way which contains the
successful features of the existing theory, and permits transitions between
equilibrium states.

				     ---

 Find an invariant torus carrying mixing flows in a generically perturbed
Hamilton system on compact manifolds! (Kolmogorov--Arnold) (This question
was the starting point for Kolmogorov which at last led to KAM theorem.
Still open.)

                                     ---

 Is there a categorial version of the KAM theory? What is the categorial and
measure status of invariant {\em mixing} torus on generally perturbed
systems? What is the right logical formulae defining the set of points
running on an invariant (mixing) torus? (O'.T.)

                                    ---
 Calculus of variation and analycity:

 Extend Hilbert's ... problem: Prove that the problem calculus of
variation for analytic functionals involving analytic
integro-differential Radon transforms admit analytic solutions.
 Solutions with positivity / convexity constraints are piecewise analytic.
 If we substitute Radon transform with Crofton transform,
then the solutions are piecewise analytic.

 If the functional is not an integral, but a {\em maximum} on values of
integro-differential Radon transforms / Crofton transforms then 
the solution is always piecewise analytic.

                                  ---

 Generalize  and give a general theory for the following problems!
Show, that the solutions are piecewise analytic and give
explicite restrictions on the number of pieces and the topology of
the analytic pieces!

 Maximize functionals (like area, width, $k^{\small th}$ outer
measure) on convex hulls of linear sets, or on soap bubbles on linear
sets with given length. Maximize functionals on linear sets
connecting a set.

 If the topology of the parameter set is allowed to vary, then they are
locally finite unions of piecewise analytic solutions.  There are explicite
restrictions on the number of pieces and the topology of the analytic
pieces.

				     ---

 Give a general theory (and investigate analyticity, by reducing the
problem using explicite restrictions to a standard one, of the
solutions) of iterated calculus of variation problems!

				     ---

 Study cooperatrive games!!!

				     ---


 Given countably many concentric sphere with radius less than one and we
know the surface area of the intersection of the surface of the body $K$ and
hyperplanes supporting the spheres. Does it determine the body?

 Let $\alpha$ be an ${\bf R}^n\times{\bf R}^m\to {\bf R}$ a non singular real
analytic function and let $\sigma_i\in {\bf R}^m$ be a countable set of
parameters. Let $S_i$ be the zero set of the $\alpha$ with
parameters $\sigma_i$ that is 
\begin{equation}
      S_i=\left\{ x\in {\bf R}^n \,:\, \alpha\(x, \sigma_i\)=0 \right\}.
\end{equation}

 Let $\beta:{\bf R}^n\times{\bf R}^n\times{\bf R}^m\to {\bf R}$ be a
positive analytic function. The $R_\beta$ Radon transform of a convex body
$K$ containing the closure of the union of the sets $S_i$ in its interior
and let $\bar R_\beta$ be its restriction to $S_i$ for $i\in {\bf N}$.

 Does $\bar R_\beta K$ determine $K$??

				     ---

 Are Pareto sets with curvature bounded below /above Alexandrov spaces?

 Is it true that if a (locally compact) Pareto set admits curvature
bounded below and above and every geodesic is locally extendable then
it is a Riemannian manifold with continuous Riemann metric?

                                  ---

 Let $M^k\subset {\bf R}^n$ be a $k$ dimensional Lipschitzian surface
and $v$ be a non constant analytic function defined in its
neighborhood. Let $U\subset M^n$ be an open neighborhood of the unit
in the isimetry group $M\(n\)$ of ${\bf R}^n$. Assume that the linear
span of the space 
 \begin{equation}\label{eq:}
     \left\{ \tau\left.v\right|_{\tau^{-1} M^k}\,:\, \tau\in U\right\}
 \end{equation}
on $M^k$ is not dense in $L^2\(M^k\)$.

 Then $k=\(n-1\)$, the manifold $M^k$ is a sphere and $v$ is a
harmonic polynomial.

 The same statement is valid for rank 1 symmetric space of non
compact type.

 The problem has close relatioships to the famous Pompeiu problem.

 In case of translations $T\(n\)$ instead of isometries $M\(n\)$, if
 \begin{equation}\label{eq:}
   \left\{ \tau\left.v\right|_{\tau^{-1} M^k}\,:\, \tau\in U\right\}
 \end{equation}
 is finite dimensional, then $k=\(n-1\)$ and $M^k$ is a sphere 
while $v$ is a a eigen function of a BVP.

 If we pose the problem in a Riemannian space with respect to small
perturbations of $M^k$ (for example by parallel vector fields) then
under mild conditions we get a characterization of balls and harmonic
functions in rank 1 Riemannian symmetric spaces.

                                  ---

 \begin{Theorem}\label{th:}
 Assume that $h$ is a harmonic function in a neighborhood of
$\Omega$ and and there exist a non zero $\lambda\in{\bf
C}^\times$ that for every $\tau\in M\(n\)$ close to the identity it
holds that
 \begin{equation}\label{eq:}
 \Delta u= \lambda u \quad{\rm on}\quad \Omega
\quad{\rm while}\quad \left.u\right|_{\tau^-1\partial\Omega}=
\quad \left.h\right|_{\tau^-1\partial\Omega}
\quad{\rm and}\quad \left.\partial_\nu u\right|_{\tau^-1\partial\Omega}
\quad \left.\partial_\nu h\right|_{\tau^-1\partial\Omega}.
 \end{equation}
 Then $\Omega$ is a ball and $h$ is a harmonic polynomial.
 \end{Theorem} 

                                  ---

 Are knots decideable?  Is it true that two knots in $S^3$ are
equivalent if complements of their tubular neighborhoods are
equivalent? Is it a knot trivial if complement of its tubular
neighborhood is a torus (that is $S^1\rimes B^2$)? Is torus
$S^1\times B^2$ decideable? Are 3 manifolds decideable??  What is the
relevance of decideability of knots to the Poincare conjecture??


				     ---

 Universal algebras [mainly for Waldhauser Tamas]:

 1. Assume that given some identities $I$. Given a clone on a set on a
finite set $X$. Can we find a finite set $Y$ containing $X$, and an algebra
on $Y$ with operations satisfying the identities $I$?

 Usually even the identities are not fixed, but only they type, and we may
ask the same questions in this type. One example for this is the so called
parameter algebras.

 2.a Characterize those varieties defined by finitely many equations, in
which finitely generated (generated by $k$ element) free algebras are a.
finite, b. have polynomial growth. Prove that the growth function is either
constant 1, (the trivial) constant $k$ (idempotent
commutative, associative) at least ${n+k-1\choose k}$ (commutative,
associative), can be estimated from the above by a polynomial of the form
$k^\alpha n^k$, or can be estimated from below by $\exp(c k^\alpha n)$,
where $\alpha>0$ is a positive constant.

 Is there a gap between $\exp(n)$ and $k^n n^n$? Can we strictly separate
algebras satisfying a non trivial identity from the free algebra, not
satisfying identity by the asymptotic behaviour of its growth function?

 2.b Characterize those finitely presented algebras which growth function is
polynomial! Is there a gap between polynomial and exponential growth? (Even
in case of groups it is open!) What can be the order growth functions?? Is
it ``continuous'' or ``discrete''?

 2.b.1 A more general question, if in the presentation we allow general
variables (like polynomial identities in groups), not only generator
elemets. This seems to be a mixed case of defining algebras by presentation
or equations (identities). Maybe, this provide a general language.

 2.c Does it help, if we assume that our algebras are topological (on a
compact / locally compact set / separable metric space / manifold) /
differentiable / algebraically defined / and admit some kind of
non-singularity / irreducibility / simplicity condition? What kind of
results we have?

 2.d One other approach is not the investigation of the growth function, but
PTIME / PSPACE reducibility / descriptivity. This may grasp better the
intuition of ``controllability'' than the growth function.

 For example: We count the number of PSPACE resource bounded equivalence
classes of terminal expressions (equivalence is provable in PSPACE), (or we
can measure the length of proof in an other way).

 We may also measure the necessary resources to a. prove equivalence, b.
enumerate all the (PSPACE resource bounded) equivalence classes, c. etc.

 In general we assume here the PTIME / PSPACE resource bounds.

 In case of more explicite bounds we may expect nice characterization of
certain identities.

 2.e All the questions are interesting, if we want ``controllability'' on
(absolute / PSPACE) random/ typical elements or on $(1-\epsilon)$ density
subsets of terminal expressions. (Maybe there are complicated things, but
with ``big probability'' the results are controllable size, and simple.)

 2.f It may be an interesting program to characterize those binary
operations which growth function is (exactly) a given function.

 2.g Some (not the asymptotic behaviour) description / characterization of
possible (exact) growth / complexity functions. (For example, by
iteration). Inequalities satisfied by growth / complexity functions.

 2.h The typical complexity of a formula is $\log (k^n n^{n-1})$. After
applying the identities (possibly in PSPACE), if it is reduced to $\log n$,
then it is of polynomial order.

 How it is related to other complexity measures?

 2.i Repeat the whole theory, if $k=n$!

 3. For every $k\in {\bf N}$ characterize those differentiable universal
algebra identities, which are locally determined by their first $k$
derivatives. (Szabo Endre, 1999)

 Different questions if the algebras are $C^k$, $C^n$ for a suitable $n>k$,
$C^\infty$, $C^\omega$. They are over ${\bf R}$, ${\bf C}$, or ${\bf Q}$.
Furthermore we can investigate the question on algebraic universal algebras
(where the algebras are defined by polinomials over an algebraically closed
field.)

 Under some of these conditions, is it true that some identities implies
others? (That is, some of these conditions are stronger than logical
strength.)

 Formally we can introduce derivatives of universal alegbras, and we can
work with formal power series expansions.

 In a more general setting, we may associate a module to an identity over an
associative ring, and we can aske the same questions formally.

 3.b Is it true that a differentiable universal algebra is always defined on
a generalized algebraic variety by generalized algebraic maps?

 3.c Is there a differentiable universal algebra ${\bf A}$ for every
$k\in{\bf N}$ that ${\cal A}$ is $k$-rigid? If not, what kind of $k$'s may
occure?

		  Number of algebras satisfying given IDs.

 Generalize the Weil conjectures for (an appropriately defined) generator
functions algebraic identities (instead of varieties).


 Approximate identities

 Given a topological / measureable universal algebra and a measure $\mu$ on
it, which behaves ``well'' with respect to the operations. Assume that the
algebra is locally connected. Then a given identity holds semi-locally with
probability 0 or 1. (Semi-local: every neighborhood of every point contains
an open set that ... .)

 Connected analytic algebras obviously admit this property.

 (Examples for the conditions: $\mu$ is a Radon measure and in any operation
of the algebra fixing except one variable, the resulting map maps $\mu$ into
a measure which is absolutely continuous with respect to $\mu$.)

 The same for categories instead of measures. Furthermore for random
elements also (in a Polish space).

 An identity holds for every random element or does not. An asymptotic
version of this statement may even hold for finite algebras.


		   Approximate identities (for checking).

 Let $I$ be an identity on $k$ variables. (For example associativity on 3
variables.) 

 For an algebra ${\bf A}$ on $X$, let $\Delta$ be the set of those $k$
tuples in $X^k$ for which $I$ does not hold. Let $\mu$ be a probabilistic
measure on $X^k$. (If $X$ is finite, let $\mu$ be the cardinality.)

 4.a (Depending on $I$) characterize the set $\Delta$.

 4.b Give estimations for $|\Delta|$ (depending on $I$).

 4.c Prove that in case of associativity, there exist a $C>0$
(independent of $|X|$) that if $|\Delta|\le C\cdot |X|^{\frac{2}{7}}$
then ${\cal A}$ is associative.

 4.d Let $\pi\(I,n\)=\min_{\left|X\right|=n}
\left\{\left|\Delta\(I,A\)\right| >0 \,:\, \right\}$.

 4.e Give estimations for $\pi\(I,n\)$, for a give $I$!

 4.f Give further characterization of $\Delta\(I,A\)$.
 So, if we are searching for algebras satisfying $I$, we do not have
to check all the identities.

 4.g Study algebras which are closest to associativity (or, in general,
to $I$) in this sense. Are they defined by finitely many identity?
Are this set of this identities computable, enumerable, etc.

 4.h The spectra of identities---the algebras not satisfying $I$ in
the $k^{\small th}$ degree.


 If the identity contains $\ge 4$ variables, the dumb check of it is getting
extremely difficult. Some probabilistic approach may help.

 Point sets on algebraic varieties.

 Result of E. Szab\'o is true
 1. With error?

 2. Over characteristic $p$?

 3. In several variables? 

 3.d When, on a metric space, we can group the results (with some error)
into polynomial many groups. This case, can we approximate the function with
some other, satisfying some identity?

 (On algebraic varieties it is more possible.)

 3.e Is there a similar result, if $F$ is a Lip continuous with known,
bounded Lip constant $L$?
 Is it true, that if a. F is analytic, b. ${\rm Lip}(F)\le L$, c.
$|F(X,Y)| < n^{2-\delta}$ (for a $\delta \ge \delta_0>0$)
then $F$ is locally a group?



			   Equations in/over groups

 1. For finite groups even minimal extensions are non unique. Give examples
for that! What can be said about minimal extensions?



			   Polynomial optimization

 Polynomial optimization on semi-algebraic varieties merging the LP and the
Gr\"obner basis algorithms. Gr\"obner bases and positivity.
(This is a problem motivated by a question of A.\ Pluh\'ar.)

 Given an algebraically defined (with positivity constraints!)
integro-differential Radon quasi-semi algebraic variation problem. Then its
solution is quasi-semi algebraic. Give (positive) Gr\"obner basis type
algorithm to compute this solutions! The same for the Pontrjagin maximum
principle!

 Using Hironaka's moethod the same for analytic integro-differential Radon
transforms with analytic positivity constraints.


