Related papers: K-trivials are NCR

We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in…

We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the…

We study the question, ``For which reals $x$ does there exist a measure $\mu$ such that $x$ is random relative to $\mu$?'' We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We…

When R is a commutative ring with identity, and if k is a natural number with kR = R, then C. Weibel proved that SK_1(R[X]) has no k-torsion. We reprove his result for any associative ring R with identity in which kR = R.

In this article, we provide an explicit constant $C$ such that there is no regular ternary sum of generalized $m$-gonal numbers for any integer $m$ greater than $C$.

We show there is no categorical metric continuum. This means that for every metric continuum X there is another metric continuum Y such that X and Y have (countable) elementarily equivalent bases but X and Y are not homeomorphic. As an…

We proved a conjecture of D. Freed that there are no non-trivial complete special Kaehler manifolds.

We describe the possible values of $K$-theory for $C(X)$ when $X$ is a co-existentially closed continuum. As a consequence we also show that all pseudo-solenoids, except perhaps the universal one, are not co-existentially closed.

It is proven, by example, that the version of $k$-means with random initialization does not have the property probabilistic k-richness.

We show that it is impossible to prove that the outcome of a quantum measurement is random.

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…

Let $X$ be a normal algebraic variety over a finitely generated field $k$ of characteristic zero, and let $\ell$ be a prime. Say that a continuous $\ell$-adic representation $\rho$ of $\pi_1^{\text{\'et}}(X_{\bar k})$ is arithmetic if there…

Given a computable probability measure P over natural numbers or infinite binary sequences, there is no computable, randomized method that can produce an arbitrarily large sample such that none of its members are outliers of P.

We shall prove that if X, Y are compact metrizable spaces of positive dimension and h: X x Y --> X is a continuous map with zero-dimensional fibers then X contains a non-trivial continuum without one-dimensional subsets; in particular X is…

Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $\xi_{ij}, i<j$ and diagonal entries $\xi_{ii}$ are independent. We show that with probability tending to 1, $M_n$ has no…

Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold…

Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability…

For any given positive integer $l$, we prove that every plane deformation of a circle which preserves the $1/2$ and $1/(2l+1)$-rational caustics is trivial i.e. the deformation consists only of similarities (rescalings plus isometries).

Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $\rho: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are…

Every K-trivial set is computable from an incomplete Martin-L\"of random set, i.e., a Martin-L\"of random set that does not compute 0'.