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1
Content available remote

Some complexity results in topology and analysis

100%
Jackson S., Daniel Mauldin R.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 1
75-83
EN
If X is a compact metric space of dimension n, then K(X), the n- dimensional kernel of X, is the union of all n-dimensional Cantor manifolds in X. Aleksandrov raised the problem of what the descriptive complexity of K(X) could be. A straightforward analysis shows that if X is an n-dimensional complete separable metric space, then K(X) is a $Σ_2^1$ or PCA set. We show (a) there is an n-dimensional continuum X in $ℝ^n+1$ for which K(X) is a complete $Π_1^1$ set. In particular, $K(X) ∈ Π_1^1-Σ_1^1$; K(X) is coanalytic but is not an analytic set and (b) there is an n-dimensional continuum X in $ℝ^n+2$ for which K(X) is a complete $Σ_2^1$ set. In particular, $K(X) ∈ Σ_2^1-Π_2^1$; K(X) is PCA, but not CPCA. It is also shown the Lebesgue measure as a function on the closed subsets of [0,1] is an explicit example of an upper semicontinuous function which is not countably continuous.
2
Content available remote

Conformal measures for rational functions revisited

81%
Denker M., Daniel Mauldin R., Nitecki Z., Urbański M.
Fundamenta Mathematicae
|
1998
|
tom 157
|
nr 2-3
161-173
EN
We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.
3
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On partitions of lines and space

81%
Erdös P., Jackson S., Daniel Mauldin R.
Fundamenta Mathematicae
|
1994
|
tom 145
|
nr 2
101-119
EN
We consider a set, L, of lines in $ℝ^n$ and a partition of L into some number of sets: $L = L_1∪...∪ L_p$. We seek a corresponding partition $ℝ^n = S_1 ∪...∪ S_p$ such that each line l in $L_i$ meets the set $S_i$ in a set whose cardinality has some fixed bound, $ω_τ$. We determine equivalences between the bounds on the size of the continuum, $2^ω ≤ ω_θ$, and some relationships between p, $ω_τ$ and $ω_θ$.
4
Content available remote

On infinite partitions of lines and space

81%
Erdös P., Jackson S., Daniel Mauldin R.
Fundamenta Mathematicae
|
1997
|
tom 152
|
nr 1
75-95
EN
Given a partition P:L → ω of the lines in $ℝ^n$, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, $Q:ℝ^n → ω$, so that each line meets at most m points of its color. Assuming Martin's Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations of these results, including to higher dimension spaces and planes.
5
Content available remote

On the Baire system generated by a linear lattice of functions

79%
Daniel Mauldin R.
Fundamenta Mathematicae
|
1970
|
tom 68
|
nr 1
51-59
6
Content available remote

Some additive properties of sets of real numbers

70%
Erdös P., Kunen K., Daniel Mauldin R.
Fundamenta Mathematicae
|
1981
|
tom 113
|
nr 3
187-199
7
Content available remote

Some examples of d-ideals and related Baire systems

69%
Daniel Mauldin R.
Fundamenta Mathematicae
|
1971
|
tom 71
|
nr 2
179-184
8
Content available remote

On rectangles and countably generated families

69%
Daniel Mauldin R.
Fundamenta Mathematicae
|
1977
|
tom 95
|
nr 2
129-139
9
Content available remote

σ-ideals and related Baire systems

60%
Daniel Mauldin R.
Fundamenta Mathematicae
|
1971
|
tom 71
|
nr 2
171-177
10
Content available remote

A representation theorem for the second dual of C[0,1]

60%
Daniel Mauldin R.
Studia Mathematica
|
1973
|
tom 46
|
nr 3
197-200
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