The Hausdorff dimension of graphs of Weierstrass functions

The Weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Though there is a simple formula for the Hausdorff dimension of the graph which is widely accepted, it has not been rigorously proved to hold. We prove that if arbitrary phases are included in each term of the summation for the Weierstrass function, the Hausdorff dimension of the graph of the function has the conjectured value for almost every sequence of phases. The argument extends to a much wider class of Weierstrass-like functions.

     

Nanotechnology in the Chemical Industry – Opportunities and Challenges

The traditional chemical industry has become a largely mature industry with many commodity products based on established technologies. Therefore, new product and market opportunities will more likely come from speciality chemicals, and from new functionalities obtained from new processing technologies as well as new microstructure control methodologies. It is a well-known fact that in addition to its molecular structure, the microstructure of a material is key to determining its properties. Controlling structures at the micro- and nano-levels is therefore essential to new discoveries. For this article, we define nanotechnology as the controlled manipulation of nanomaterials with at least one dimension less than 100nm. Nanotechnology is emerging as one of the principal areas of investigation that is integrating chemistry and materials science, and in some cases integrating these with biology to create new and yet undiscovered properties that can be exploited to gain new market opportunities. In this article market opportunities for nanotechnology will be presented from an industrial perspective covering electronic, biomedical, performance materials, and consumer products. Manufacturing technology challenges will be identified, including operations ranging from particle formation, coating, dispersion, to characterization, modeling, and simulation. Finally, a nanotechnology innovation roadmap is proposed wherein the interplay between the development of nanoscale building blocks, product design, process design, and value chain integration is identified. A suggestion is made for an R&D model combining market pull and technology push as a way to quickly exploit the advantages in nanotechnology and translate these into customer benefits.     

Cotangent cohomology of Stanley-Reisner rings

Simplicial complexes X provide commutative rings A(X) via the Stanley- Reisner construction. We calculated the cotangent cohomology, i.e., T¹ and T² of A(X) in terms of X. These modules provide information about the deformation theory of the algebro geometric objects assigned to X.     

Geometric algebra techniques for general relativity

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.     

High-resolution infrared spectra of NCCN and NCCN

High-resolution spectra of ¹⁵N¹²C¹²C¹⁵N and ¹⁴N¹³C¹³C¹⁴N have been measured and analyzed from 200 to 3600 cm⁻¹. All the vibrational levels below 900 cm⁻¹ have been observed and characterized. The Fermi resonance between ν₂ and 2ν₄ has been studied and the resonance constant has been determined for several cases. Several Σ⁻ states have been directly observed for the first time for each isotopomer, the (0001111)0f, (0011111)0f, and (0002222)0f states. The pattern of the energy levels for clusters of l-type resonance coupled levels, such as 0001131,3, has been determined for cyanogen for the first time. Among other things this involved the determination of the vibrational l-type resonance constant, r₄₅. Many of the power series constants, α_i and x_ij, and higher order constants have been determined.     

Analysis of Antennas with PBG Substrate

The use of photonic materials has been used in the theory of optical waves. The PBG (Photonic Band Gap) theory and material, was developed recently for optical frequencies and can be easily applied to millimeter waves, microwaves and planar antenna frequencies. The presence of photonic materials as substrate in antennas has some good characteristics such as, supression of light spontaneous emission and suppression of surface waves, allowing the application in planar antenna array. In this work an elaborate analysis using the full wave Transverse Transmission Line - TTL method, that provides efficient and concise results is applied to the planar antennas array with PBG substrate.     

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