Homology of Distributive Lattices

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology of a finite distributive lattice. We also indicate the answer for unital spindles and conjecture the general formula for semi-lattices and some skew lattices. Then we propose a generalization of a lattice as a set with a number of idempotent operations satisfying the absorption law.Comment: 30 pages, 3 tables, 3 figure

Nondestructive monitoring of ageing of Alkali resistant Glass fiber reinforced cement (GRC)

Glass fiber reinforced cement (GRC) is a composite material made of portland cement mortar and alkali resistant (AR) fibers. AR fibers are added to portland cement to give the material additional flexural strength and toughness. However, ageing deteriorates the fibers and as a result the improvement in the mechanical properties resulted from the fiber addition disappears as the structure becomes old. The aim of this paper is monitoring GRC ageing by nondestructive evaluation (NDE) techniques. Two different NDE techniques (1) nonlinear impact resonant acoustic spectroscopy analysis and (2) propagating ultrasonic guided waves are used for this purpose. Both techniques revealed a reduction of the nonlinear behavior in the GRC material with ageing. Specimens are then loaded to failure to obtain their strength and stiffness. Compared to the un-aged specimens, the aged specimens are found to exhibit more linear behavior, have more stiffness but less toughness. Finally, undisturbed fragments on the fracture surface from mechanical tests are inspected under the electron microscope, to understand the fundamental mechanisms that cause the change in the GRC behavior with ageing.The authors want to acknowledge the financial support of the Ministerio de Ciencia e Innovacion MICINN, Spain, and FEDER funding (Ondacem Project: BIA 2010-19933) and BES-2011-044624. Also thanks to PAID-02-11 Program from Universitat Politecnica de Valencia.Eiras Fernández, JN.; Kundu, T.; Bonilla Salvador, MM.; Paya Bernabeu, JJ. (2013). Nondestructive monitoring of ageing of Alkali resistant Glass fiber reinforced cement (GRC). Journal of Nondestructive Evaluation - NDT and E International. 32:300-314. https://doi.org/10.1007/s10921-013-0183-yS30031432Bentur, A., Fibre, M.S.: Reinforced Cementitious Composites, 2nd edn. Taylor and Francis, New York (2007)Purnell, P., Short, N.R., Page, C.L.: A static fatigue model for the durability of glass fibre reinforced cement. J. 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Thickness of knots

AbstractClassical knot theory studies one-dimensional filaments; in this paper we model knots as more physically “real”, e.g., made of some “rope” with nonzero thickness. A motivating question is: How much length of unit radius rope is needed to tie a nontrivial knot?For a smooth knot K, the “injectivity radius” R(K) is the supremum of radii of embedded tubular neighborhoods. The “thickness” of K, a new measure of knot complexity, is the ratio of R(K) to arc-length. We relate thickness to curvature, self-distance, distortion, and (for knot types) edge-number

Thickness Of Knots

this paper we study physical knots; that is, knots tied (as closed loops) in real pieces of rope, which have diameter. Intuitively, for a given diameter, one needs a certain minimum length of rope in order to tie a (non-trivial) knot, and (more vaguely), the more complicated the knot you want to tie, the more rope you need. To be specific, we can ask