We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any RCD spaces with the Bishop-type ...inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such RCD^*(K,N) spaces is exactly N. We also prove that every tangent cone at any point on such RCD spaces is a metric cone.
This paper contains two new projection theorems in the plane.
First, let K⊂B(0,1)⊂R2 be a set with H∞1(K)∼1, and write πe(K) for the orthogonal projection of K into the line spanned by e∈S1. For ...1/2≤s<1, writeEs:={e:N(πe(K),δ)≤δ−s}, where N(A,r) is the r-covering number of the set A. It is well-known – and essentially due to R. Kaufman – that N(Es,δ)⪅δ−s. Using the polynomial method, I prove thatN(Es,r)⪅min{δ−s(δr)1/2,r−1},δ≤r≤1. I construct examples showing that the exponents in the bound are sharp for δ≤r≤δs.
The second theorem concerns projections of 1-Ahlfors–David regular sets. Let A≥1 and 1/2≤s<1 be given. I prove that, for p=p(A,s)∈N large enough, the finite set of unit vectors Sp:={e2πik/p:0≤k<p} has the following property. If K⊂B(0,1) is non-empty and 1-Ahlfors–David regular with regularity constant at most A, then1p∑e∈SpN(πe(K),δ)≥δ−s for all small enough δ>0. In particular, dim‾Bπe(K)≥s for some e∈Sp.
We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in Rd under two simple geometric ...conditions: The Dirichlet boundary part is Ahlfors–David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regularity results for the fractional powers of the Laplacian with boundary conditions in our geometric setup.
Let
A
,
B
⊂
R
be closed Ahlfors-regular sets with dimensions
dim
H
A
=
:
α
and
dim
H
B
=
:
β
. I prove that
dim
H
A
+
θ
B
≥
α
+
β
·
1
-
α
2
-
α
for all
θ
∈
R
\
E
, where
dim
H
E
=
0
.
Efficient domination through eigenvalues Cardoso, Domingos M.; Lozin, Vadim V.; Luz, Carlos J. ...
Discrete Applied Mathematics,
12/2016, Letnik:
214
Journal Article
Recenzirano
Odprti dostop
The paper begins with a new characterization of (k,τ)-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or ...not a graph contains a (0,τ)-regular set. When τ=1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NP-complete. If −1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it does not work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs.
In this paper, we consider some classes of general Cantor sets satisfying (αk)- regular condition and obtain necessary and sufficient conditions to characterize the fatness and thinness of the sets ...for doubling measures under suitable conditions. It is applied to classify some kinds of Cantor-type sets.
Let Ω ⊂ ℝ𝑛⁺¹,𝑛 ≥ 2, be a 1-sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that 𝜕Ω is n-dimensional ...Ahlfors-David regular. We characterize the rectifiability of 𝜕Ω in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that 𝜕Ω can be covered ℋ𝑛-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of Ω and to the fact that 𝜕Ω possesses exterior corkscrew points in a qualitative way ℋ𝑛-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.
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