#### Filter Results:

- Full text PDF available (21)

#### Publication Year

2005

2014

- This year (0)
- Last 5 years (2)
- Last 10 years (13)

#### Publication Type

#### Co-author

#### Journals and Conferences

#### Key Phrases

Learn More

3. (a) If F is as in Theorem 10.7, put A = F′(0), F1(x) = A −1F(x). Then F1(0) = I. Show that F1(x) = Gn◦Gn−1◦. . .◦G1(x) in some neighbourhood of 0, for certain primitive mappings G1, . . . ,Gn. This gives another version of Theorem 10.7: F(x) = F(0)Gn◦Gn−1◦. . .◦G1(x). (b) Prove that the mapping (x, y) 7→ (y, x) of R onto R is not the composition of any… (More)

Tube formulas (by which we mean an explicit formula for the volume of an (inner) ε-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl’s celebrated results on spectral asymptotics, and the subsequent relation between… (More)

A resistance network is a connected graph (G, c) with edges (and edge weights) determined by the conductance function cxy . The Dirichlet energy form E produces a Hilbert space structure (which we call the energy space HE) on the space of functions of finite energy. In a previous paper, we constructed a reproducing kernel {vx} for this Hilbert space and… (More)

A resistance network is a connected graph (G, c). The conductance function cxy weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form E produces a Hilbert space structure (which we call the energy space HE) on the space of functions of finite energy. We use the reproducing kernel {vx} constructed… (More)

A formula for the interior ε-neighbourhood of the classical von Koch snowflake curve is computed in detail. This function of ε is shown to match quite closely with earlier predictions from [La-vF1] of what it should be, but is also much more precise. The resulting ‘tube formula’ is expressed in terms of the Fourier coefficients of a suitable nonlinear and… (More)

We extend some aspects of the theory of fractal strings and their complex dimensions from the real line to general Euclidean spaces. This is accomplished by using the explicit formulas and techniques of [La-vF4]. We use the self-similar tilings constructed in [Pe1] to define a zeta function which encodes the scaling properties of the tiling. This allows us… (More)

An iterated function system Φ consisting of contractive affine mappings has a unique attractor F ⊆ R which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling T of the convex hull of the attractor. These tiles form a collection of sets whose… (More)

An electrical resistance network (ERN) is a connected graph (G, c). The conductance function cxy weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form E and the discrete Laplace operator ∆ on a finite network is given by E(u, v) = 〈u,∆v〉2 , where the latter is the usual… (More)

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely,… (More)

We study the boundary theory of a connected weighted graph G from the viewpoint of stochastic integration. For the Hilbert space HE of Dirichlet-finite functions on G, we construct a Gel’fand triple S ⊆ HE ⊆ S′. This yields a probability measure P on S′ and an isometric embedding of HE into L2(S′,P), and hence gives a concrete representation of the boundary… (More)