Table of Contents

Abstract: A review of the theory describing the coexistence between d-wave superconductivity and s-wave charge-density-waves (CDWs) is presented. The CDW gapping is identified with pseudogapping observed in high-T_c oxides. According to the cuprate specificity, the analysis is carried out for the two-dimensional geometry of the Fermi surface (FS). Phase diagrams on the σ₀ − α plane—here, σ₀ is the ratio between the energy gaps in the parent pure CDW and superconducting states, and the quantity 2α is connected with the degree of dielectric (CDW) FS gapping—were obtained for various possible configurations of the order parameters in the momentum space. Relevant tunnel and photoemission experimental data for high-T_c oxides are compared with theoretical predictions. A brief review of the results obtained earlier for the coexistence between s-wave superconductivity and CDWs is also given.

Abstract: The language of gene expression displays topological symmetry. An important step during gene expression is the binding of transcriptional proteins to DNA promoters adjacent to a gene. Some proteins bind to many promoters in a genome, defining a regulon of genes wherein each promoter might vary in DNA sequence relative to the average consensus. Here we examine the linguistic organization of gene promoter networks, wherein each node in the network represents a promoter and links between nodes represent the extent of base pair-sharing. Prior work revealed a fractal nucleus in several σ-factor regulons from Escherichia coli. We extend these findings to show fractal nuclei in gene promoter networks from three bacterial species, E. coli, Bacillus subtilis, and Pseudomonas aeruginosa. We surveyed several non-σ transcription factors from these species and found that many contain a nucleus that is both visually and numerically fractal. Promoter footprint size scaled as a negative power-law with both information entropy and fractal dimension, while the latter two parameters scaled positively and linearly. The fractal dimension of the diffuse networks (d_B = ~1.7) was close to that expected of a diffusion limited aggregation process, confirming prior predictions as to a possible mechanism for development of this structure.

Abstract: The paper puts the emphasis on surveying information-theoretic network measures for analyzing the structure of networks. In order to apply the quantities interdisciplinarily, we also discuss some of their properties such as their structural interpretation and uniqueness.

Abstract: In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N²_a 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La₂CuO₄ is shortly reviewed.

Abstract: We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m, or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains. We display the algorithms’ output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3].