zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Spectral decimation on Hambly’s homogeneous hierarchical gaskets. (English) Zbl 1211.28005

Using the so-called “spectral decimation”, the authors establish a frame on Hambly’s homogeneous hierarchical gaskets. There are 5 sections in this paper:

1. Introduction – some definitions are given: a hierarchical fractal; the concept of finitely ramified: the homogeneous hierarchical gasket, denoted by HH(b); a fractal Γ realized as a limit of a sequence of graphs Γ 0 ,Γ 1 , with vertices V 0 V 1 . Take V 0 ={q 0 ,q 2 ,q 2 }, as vertices of a triangle, considered as the boundary of a HH(b), then the un-renormalized energy E m (u) of a function on V m ; renormalized energy 𝔼 m (u); energy on HH(b); E(u)=lim m 𝔼 m (u); a bilinear form E(u,v) as well as the standard Laplacian Δu.

2. The spectral decimation on SG 3 , the usual Sierpiński gasket.

3. Dirichlet and Neumann spectra for SG 3 .

4. Spectral decimation on homogeneous hierarchical gaskets.

5. Spectral gaps.

As applications, the paper shows that these spectra have infinitely many large spectral gaps. And under certain restrictions, a computer-assisted proof that the set of ratios of eigenvalues has gaps, implying the existence of quasi-elliptic PDE’s on the product of two such fractals.

MSC:
28A80Fractals
31C99Generalizations in potential theory
35H99Close-to-elliptic equations