Networks of myelin covariance.

Networks of anatomical covariance have been widely used to study connectivity patterns in both normal and pathological brains based on the concurrent changes of morphometric measures (i.e., cortical thickness) between brain structures across subjects (Evans, ). However, the existence of networks of microstructural changes within brain tissue has been largely unexplored so far. In this article, we studied in vivo the concurrent myelination processes among brain anatomical structures that gathered together emerge to form nonrandom networks. We name these "networks of myelin covariance" (Myelin-Nets). The Myelin-Nets were built from quantitative Magnetization Transfer data-an in-vivo magnetic resonance imaging (MRI) marker of myelin content. The synchronicity of the variations in myelin content between anatomical regions was measured by computing the Pearson's correlation coefficient. We were especially interested in elucidating the effect of age on the topological organization of the Myelin-Nets. We therefore selected two age groups: Young-Age (20-31 years old) and Old-Age (60-71 years old) and a pool of participants from 48 to 87 years old for a Myelin-Nets aging trajectory study. We found that the topological organization of the Myelin-Nets is strongly shaped by aging processes. The global myelin correlation strength, between homologous regions and locally in different brain lobes, showed a significant dependence on age. Interestingly, we also showed that the aging process modulates the resilience of the Myelin-Nets to damage of principal network structures. In summary, this work sheds light on the organizational principles driving myelination and myelin degeneration in brain gray matter and how such patterns are modulated by aging

Topological modular forms and conformal nets

We describe the role conformal nets, a mathematical model for conformal field theory, could play in a geometric definition of the generalized cohomology theory TMF of topological modular forms. Inspired by work of Segal and Stolz-Teichner, we speculate that bundles of boundary conditions for the net of free fermions will be the basic underlying objects representing TMF-cohomology classes. String structures, which are the fundamental orientations for TMF-cohomology, can be encoded by defects between free fermions, and we construct the bundle of fermionic boundary conditions for the TMF-Euler class of a string vector bundle. We conjecture that the free fermion net exhibits an algebraic periodicity corresponding to the 576-fold cohomological periodicity of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we establish a lower bound of 24 on this periodicity of the free fermions

Experimental investigation of the mechanical stiffness of periodic framework-patterned elastomers

Recent advances in the cataloguing of three-dimensional nets mean a systematic search for framework structures with specific properties is now feasible. Theoretical arguments about the elastic deformation of frameworks suggest characteristics of mechanically isotropic networks. We explore these concepts on both isotropic and anisotropic networks by manufacturing porous elastomers with three different periodic net geometries. The blocks of patterned elastomers are subjected to a range of mechanical tests to determine the dependence of elastic moduli on geometric and topological parameters. We report results from axial compression experiments, three-dimensional X-ray computed tomography imaging and image-based finite-element simulations of elastic properties of framework-patterned elastomers

Compactness Determines the Success of Cube and Octahedron Self-Assembly

Nature utilizes self-assembly to fabricate structures on length scales ranging from the atomic to the macro scale. Self-assembly has emerged as a paradigm in engineering that enables the highly parallel fabrication of complex, and often three-dimensional, structures from basic building blocks. Although there have been several demonstrations of this self-assembly fabrication process, rules that govern a priori design, yield and defect tolerance remain unknown. In this paper, we have designed the first model experimental system for systematically analyzing the influence of geometry on the self-assembly of 200 and 500 µm cubes and octahedra from tethered, multi-component, two-dimensional (2D) nets. We examined the self-assembly of all eleven 2D nets that can fold into cubes and octahedra, and we observed striking correlations between the compactness of the nets and the success of the assembly. Two measures of compactness were used for the nets: the number of vertex or topological connections and the radius of gyration. The success of the self-assembly process was determined by measuring the yield and classifying the defects. Our observation of increased self-assembly success with decreased radius of gyration and increased topological connectivity resembles theoretical models that describe the role of compactness in protein folding. Because of the differences in size and scale between our system and the protein folding system, we postulate that this hypothesis may be more universal to self-assembling systems in general. Apart from being intellectually intriguing, the findings could enable the assembly of more complicated polyhedral structures (e.g. dodecahedra) by allowing a priori selection of a net that might self-assemble with high yields

Beyond Topologies, Part I

Arguments on the need, and usefulness, of going beyond the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are presented. The motivation comes, among others, from well known {\it topological type processes}, or in short TTP-s, in the theories of Measure, Integration and Ordered Spaces. These TTP-s, as shown by the classical characterization given by the {\it four Moore-Smith conditions}, can {\it no longer} be incorporated within the usual HKB topologies. One of the most successful recent ways to go beyond HKB topologies is that developed in Beattie & Butzmann. It is shown in this work how that extended concept of topology is a {\it particular} case of the earlier one suggested and used by the first author in the study of generalized solutions of large classes of nonlinear partial differential equations

Survey of mathematical foundations of QFT and perturbative string theory

Recent years have seen noteworthy progress in the mathematical formulation of quantum field theory and perturbative string theory. We give a brief survey of these developments. It serves as an introduction to the more detailed collection "Mathematical Foundations of Quantum Field Theory and Perturbative String Theory".Comment: This is the introduction to the upcoming volume "Mathematical Foundations of Quantum Field Theory and Perturbative String Theory", edited by the authors and published by the American Mathematical Societ