Şahin Koçak

TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

A note on positive Lyapunov exponent and sensitive dependence on initial conditions

Abstract In this note, we give two examples of discrete dynamics (functions) on intervals ⊂ R , the first having a positive Lyapunov exponent at a point but not being sensitive-dependent on initial conditions at that point; and the second being sensitive-dependent on initial conditions at a point but having a negative Lyapunov exponent at that point.

Monopole Equations on 8-Manifolds with Spin(7) Holonomy

We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit non-trivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case.

The geometry of self-dual two-forms

We show that self-dual two-forms in 2n-dimensional spaces determine a n2−n+1-dimensional manifold S2n and the dimension of the maximal linear subspaces of S2n is equal to the (Radon–Hurwitz) number of linearly independent vector fields on the sphere S2n−1. We provide a direct proof that for n odd S2n has only one-dimensional linear submanifolds. We exhibit 2c−1-dimensional subspaces in dimensions which are multiples of 2c, for c=1,2,3. In particular, we demonstrate that the seven-dimensional linear subspaces of S8 also include among many other interesting classes of self-dual two-forms, the self-dual two-forms of Corrigan, Devchand, Fairlie, and Nuyts [Nucl. Phys. B 214, 452 (1983)] and a representation of Cl7 given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras.

Self-dual Yang-Mills fields in eight dimensions

AbstractStrongly self-dual Yang-Mills fields in even-dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fieldsFμν. We derive a topological bound on R8,

On Products of Uniquely Geodesic Spaces

\int_M {\left( {F, F} \right)} ^2 \geqslant k \int_M {p_{\text{1}}^{\text{2}} }

Canonical bases for real representations of Clifford algebras

, wherep1 is the first Pontryagin class of the SO(n) Yang-Mills bundle, andk is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields

Fractal interpolation on the Sierpinski Gasket

F_{\mu \nu}