\def\({\left(} \def\){\right)} Problems for thinking % \baselineskip = 2\baselineskip 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.