43 research outputs found
All functions g:N-->N which have a single-fold Diophantine representation are dominated by a limit-computable function f:N\{0}-->N which is implemented in MuPAD and whose computability is an open problem
Let E_n={x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For any
integer n \geq 2214, we define a system T \subseteq E_n which has a unique
integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are
positive and max(a_1,...,a_n)>2^(2^n). For a positive integer n, let f(n)
denote the smallest non-negative integer b such that for each system S
\subseteq E_n with a unique solution in non-negative integers x_1,...,x_n, this
solution belongs to [0,b]^n. We prove that if a function g:N-->N has a
single-fold Diophantine representation, then f dominates g. We present a MuPAD
code which takes as input a positive integer n, performs an infinite loop,
returns a non-negative integer on each iteration, and returns f(n) on each
sufficiently high iteration.Comment: 17 pages, Theorem 3 added. arXiv admin note: substantial text overlap
with arXiv:1309.2605. text overlap with arXiv:1404.5975, arXiv:1310.536
Iterated relative recursive enumerability
A result of Soare and Stob asserts that for any non-recursive r.e. set C , there exists a r.e.[ C ] set A such that A ⊕ C is not of r.e. degree. A set Y is called [of] m -REA ( m -REA[ C ] [degree] iff it is [Turing equivalent to] the result of applying m -many iterated ‘hops’ to the empty set (to C ), where a hop is any function of the form X → X ⊕ W e X . The cited result is the special case m =0, n =1 of our Theorem. For m =0,1, and any ( m +1)-REA set C , if C is not of m -REA degree, then for all n there exists a n -r.e.[ C ] set A such that A ⊕ C is not of ( m+n )-REA degree. We conjecture that this holds also for m ≥2.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46068/1/153_2005_Article_BF01278463.pd
Universal fluctuations in subdiffusive transport
Subdiffusive transport in tilted washboard potentials is studied within the
fractional Fokker-Planck equation approach, using the associated continuous
time random walk (CTRW) framework. The scaled subvelocity is shown to obey a
universal law, assuming the form of a stationary Levy-stable distribution. The
latter is defined by the index of subdiffusion alpha and the mean subvelocity
only, but interestingly depends neither on the bias strength nor on the
specific form of the potential. These scaled, universal subvelocity
fluctuations emerge due to the weak ergodicity breaking and are vanishing in
the limit of normal diffusion. The results of the analytical heuristic theory
are corroborated by Monte Carlo simulations of the underlying CTRW
Constructive Dimension and Turing Degrees
This paper examines the constructive Hausdorff and packing dimensions of
Turing degrees. The main result is that every infinite sequence S with
constructive Hausdorff dimension dim_H(S) and constructive packing dimension
dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) /
dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0,
then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness
extractor* that increases the algorithmic randomness of S, as measured by
constructive dimension.
A number of applications of this result shed new light on the constructive
dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to
hold for the Turing degree of any sequence S. A new proof is given of a
previously-known zero-one law for the constructive packing dimension of Turing
degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) =
dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive
Hausdorff and packing dimension equal to 1.
Finally, it is shown that no single Turing reduction can be a universal
constructive Hausdorff dimension extractor, and that bounded Turing reductions
cannot extract constructive Hausdorff dimension. We also exhibit sequences on
which weak truth-table and bounded Turing reductions differ in their ability to
extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems,
45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to
insufficient care with the choice of delta. This version modifies that proof
to fix the error
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure