Abstract The search for novel topological quantum states has recently moved beyond naturally occurring crystalline materials to complex and engineered systems. In this work we generalize the notion of electronic random lattices in non-integer dimensions. By considering a class D tight-binding model on critical clusters resulting from two-dimensional site percolation process, demonstrate that th...

This paper investigates an approach to begin the study of fractals in architectural design. Vector-based fractals are studied to determine if the modification of vector direction in either the generator or the initiator will develop alternate fractal forms. The Koch Snowflake is used as the demonstrating fractal. Introduction A fractal is an object or quantity that displays self-similarity on a...

It is possible for one to define fractals as those sets which have non-integral Hausdorff Dimension. This paper defines Hausdorff Dimension and rigorously introduces the necessary theory to prove that one such set is indeed a fractal. The first chapter includes a proof of Banach’s Fixed Point Theorem. The Hausdorff Metric is defined and it is mentioned how one produces ‘generators’ for creating...

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to product of fractals. While our resul...

Abstract— Fractal based CBIR is based on the self similarity fundamentals of fractals. Mathematical and natural fractals are the shapes whose roughness and fragmentation neither tend to vanish, nor fluctuate, but remain essentially unchanged as one zooms in continually and examination is refined. Since an image can be characterized by its fractal code, a fractal code can therefore be used as a ...

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our resu...