The Solution to the Navier Stokes Problem: A Topological Approach

We introduce a geometric and spectral framework for the incompressible Navier-Stokes equations on the Hopf fibration S^1 -\u3e S^3 -\u3e CP^1, and apply it to study aspects of the Clay Millennium Prize problem on global regularity. In real time (t in R), we establish a non-closure theorem for triadic interactions, showing that the combined base and fiber spectra {k(k+1)} and {m^2} are arithmetically incommensurate, leading to proliferation of an unbounded set from initial configurations with at least two base and two fiber indices. Employing Littlewood-Paley analysis, a helical Beltrami decomposition, representation-theoretic coupling on S^3 ~= SU(2), and bounds on oscillatory phases, we derive a flux coercivity theorem at dyadic frontiers. This yields a frontier bootstrap argument and a conditional blowup theorem under a generic homochiral bias. In complex time (t in C with Re(t) \u3e 0), we demonstrate global existence and regularity for initial data in critical spaces via analytic semigroup techniques. Additionally, we establish an exact equivalence between the classical Navier-Stokes equations on R^3 and a conformally covariant formulation on S^3 \ {N} via stereographic projection, specifying the requisite weights and pressure adjustments for bijective solution correspondence. The results highlight a dichotomy: real-time evolution encounters structural obstructions due to spectral incommensurability, while complex-time dynamics enables global solvability. WRITTEN WITH INTENT OF SUBMISSION TO PEER REVIEWED JOURNAL FOR CONSIDERATION FOR THE CLAY-MILLENIUM PRIZE. Note : LaTex errors noted to be correcte

Formation of aggregated nanoparticle spheres through femtosecond laser surface processing

A detailed structural and chemical analysis of a class of self-organized surface structures, termed aggregated nanoparticle spheres (AN-spheres), created using femtosecond laser surface processing (FLSP) on silicon, silicon carbide, and aluminum is reported in this paper. AN-spheres are spherical microstructures that are 20–100 μm in diameter and are composed entirely of nanoparticles produced during femtosecond laser ablation of material. AN-spheres have an onion-like layered morphology resulting from the build-up of nanoparticle layers over multiple passes of the laser beam. The material properties and chemical composition of the AN-spheres are presented in this paper based on scanning electron microscopy (SEM), focused ion beam (FIB) milling, transmission electron microscopy (TEM), and energy dispersive x-ray spectroscopy (EDX) analysis. There is a distinct difference in the density of nanoparticles between concentric rings of the onion-like morphology of the AN-sphere. Layers of high-density form when the laser sinters nanoparticles together and low-density layers form when nanoparticles redeposit while the laser ablates areas surrounding the AN-sphere. The dynamic nature of femtosecond laser ablation creates a variety of nanoparticles that make-up the AN-spheres including Si/C core-shell, nanoparticles that directly fragmented from the base material, nanoparticles with carbon shells that retarded oxidation, and amorphous, fully oxidized nanoparticles

Micro/nanostructures formation by femtosecond laser surface processing on amorphous and polycrystalline Ni60Nb40

Femtosecond laser surface processing is a technology that can be used to functionalize many surfaces, imparting specialized properties such as increased broadband optical absorption or superhydrophilicity/superhydrophobicity. In this study, two unique classes of surface structures, below surface growth (BSG) and above surface growth (ASG) mounds, were formed by femtosecond laser surface processing on amorphous and polycrystalline Ni60Nb40 with two different grain sizes. Cross sectional imaging of these mounds revealed thermal evidence of the unique formation processes for each class of surface structure. BSG mounds formed on all three substrates using the same laser parameters had similar surface morphology. The microstructures in the mounds were unaltered compared with the substrate before laser processing, suggesting their formation was dominated by preferential valley ablation. ASG mounds had similar morphology when formed on the polycrystalline Ni60Nb40 substrates with 100 nm and 2 [H9262]m grain size. However, the ASG mounds had significantly wider diameter and higher peak-to-valley heights when the substrate was amorphous Ni60Nb40. Hydrodynamic melting was primarily responsible for ASG mound formation. On amorphous Ni60Nb40 substrates, the ASG mounds are most likely larger due to lower thermal diffusivity. There was clear difference in growth mechanism of femtosecond laser processed BSG and ASG mounds, and grain size does not appear to be a factor

A cellular model for studying accommodation to environmental stressors: Protection and potentiation by cadmium and other metals

Exposure of P. polycephalum to a subthreshold challenge of Cd2+, which did not delay mitosis, elicited a protective response against a mitotic delay resulting from subsequent exposure to a suprathreshold challenge of Cd2+. Some characteristics of this protective response are herein identified. The concentration of Cd2+ in the subthreshold challenge could be lowered to 10-5 m and maintain complete protection against a suprathreshold challenge of 5 x 10-4 m Cd2+. A subthreshold challenge of 10-4 m provided full protection against a Cd2+ concentration of 7 x 10-4 m. A subthreshold challenge of 10-4 m Cd2+ could be placed anywhere in the cell cycle approaching but not abutting a suprathreshold challenge of 5 x 10-4 m Cd2+ in late G2 and still provide complete protection with the exception of one point in early S. At that point, 10-4 m Cd2+ by itself was toxic to the cell; partial protection, however, developed. Other responses developed when metals were substituted for cadmium. Cd2+ protected against exposure to Hg2+ and Ni2+ but potentiated exposures to Co2+, Cu2+, Pb2+, or Zn2+. A curious observ ation is that exposure to Hg2+ and Ni2+ protected against exposure to Cd2+, while exposure to Co2+, Pb2+, or Zn2+ potentiated exposure to Cd2+. Hg2+ and Ni2+ protected against reexposure to themselves.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22746/1/0000301.pd

A cellular model for studying accommodation to environmental stressors: A protective response to subtoxic exposure to cadmium

A model is described for testing the effect of exposure to subtoxic challenge upon cellular integrity. The model incorporates Physarum polycephalum as a biological assay system, the ability of the cell to traverse the cell cycle as an indicator of cell integrity, and the use of repeated challenge by cadmium ion as a mechanism for amplifying the response to subthreshold exposure. A sensitivity profile of Physarum, developed by periodic exposure to 5 x 10-4 m Cd2+ for 30 min throughout the cell cycle, contains two peaks of sensitivity resulting in mitotic delay, one in early S and the other in late G2. Physarum accommodates to a subtoxic challenge of Cd2+ by developing a protective response: Exposure to 10-4 m Cd2+ for 30 min in early G2 (0.45 cycle), which does not delay mitosis, protects Physarum against a mitotic delay of 105 min resulting from exposure to 4 x 10-4 m Cd2+ for 30 min in late G2 (0.75 cycle). Protection persists for at least two cell cycles.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22745/1/0000300.pd

Microstructure and Phase Analysis in Mn-Al and Zr-Co Permanent Magnets

In America’s search for energy independence, the development of rare-earth free permanent magnets is one hurdle that still stands in the way. Permanent magnet motors provide a higher efficiency than induction motors in applications such as hybrid vehicles and wind turbines. This thesis investigates the ability of two materials, Mn-Al and Zr-Co, to fill this need for a permanent magnet material whose components are readily available within the U.S. and whose supply chain is more stable than that of the rare-earth materials. This thesis focuses on the creation and optimization of these two materials to later be used as the hard phase in nanocomposites with high energy products (greater than 10 MGOe). Mn-Al is capable of forming the pure L10 structure at a composition of Mn54Al43C3. When Mn is replaced by Fe or Cu using the formula Mn48Al43C3T6 the anisotropy constant is lowered from 1.3∙107 ergs/cm3 to 1.0∙107 ergs/cm3 and 0.8∙107 ergs/cm3 respectively. Previous studies have reported a loss in magnetization in Mn-Al alloys during mechanical milling. The reason for this loss in magnetization was investigated and found to be due to the formation of the equilibrium β-Mn phase of the composition Mn3Al2 and not due to oxidation or site disorder. It was also shown that fully dense Mn-Al permanent magnets can be created at hot pressing temperatures at or above 700oC and that the ε-phase to τ-phase transition and consolidation can be combined into a single processing step. The addition of small amounts of Cu to the alloy, 3% atomic, can increase the compaction density allowing high densities to be achieved at lower pressing temperatures. While the structure is still under debate, alloys at the composition Zr2Co11 in the Zr-Co system have been shown to have hard magnetic properties. This thesis shows that multiple structures exist at this Zr2Co11 composition and that altering the cooling rate during solidification of the alloy affects the ratio of the phase composition and therefore affects the magnetic properties. Phase diagrams for the Zr-Co system show that the Zr2Co11 phase is stable to a temperature of 1272oC, at which point the Zr6Co23 phase is the most favorable. However, this thesis shows that the Zr6Co23 phase forms at room temperature during high energy mechanical milling and at annealing temperatures as low as 600oC. Since high energy mechanical milling was not a potential method to creating single crystalline particles, hydrogen embrittlement was investigated. Hydrochloric acid was used to induce hydrogen embrittlement in the Zr2Co11 alloy, modifying the fracture characteristics of the alloy causing it to occur primarily along grain boundaries resulting in single crystalline particles with remanent magnetization enhancement. Adviser: Jeffrey E. Shiel