Uniform density in matroids, matrices and graphs

We give new characterizations for the class of uniformly dense matroids, and we describe applications to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope contains a point with constant coordinates, and if and only if there exists a measure on the bases such that every element of the ground set has equal probability to be in a random basis with respect to this measure. As one application, we derive new spectral, structural and classification results for uniformly dense graphic matroids. In particular, we show that connected regular uniformly dense graphs are $1$-tough and thus contain a (near-)perfect matching. As a second application, we show that strictly uniformly dense real representable matroids can be represented by projection matrices with constant diagonal and that they are parametrized by a subvariety of the real Grassmannian.Comment: 23 page

Revisiting path-type covering and partitioning problems

This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts

Asymptotic Freedom: From Paradox to Paradigm

Asymptotic freedom was developed as a response to two paradoxes: the weirdness of quarks, and in particular their failure to radiate copiously when struck; and the coexistence of special relativity and quantum theory, despite the apparent singularity of quantum field theory. It resolved these paradoxes, and catalyzed the development of several modern paradigms: the hard reality of quarks and gluons, the origin of mass from energy, the simplicity of the early universe, and the power of symmetry as a guide to physical law.Comment: 26 pages, 10 figures. Lecture on receipt of the 2004 Nobel Prize. v2: typo (in Ohm's law) correcte

Mechanical and self-sensing properties of concrete reinforced with carbon nanofibres

Concrete is extensively used in the construction industry, but the formation and development of cracks undermines the integrity of concrete structures. Improving the mechanical properties of concrete and developing efficient health monitoring of structures are thus essential tasks to be tackled. The research described in this paper was concerned with the effect of nanofibres on the mechanical properties of concrete as the use of nanofibres such as carbon nanofibres (CNFs) within cementitious materials has been found to be effective in enhancing the mechanical properties of concrete as well as its sensing ability. Most previous works have focused on evaluating the microstructure and mechanical behaviour of nano-reinforced mortar. Only a few studies have attempted to evaluate the mechanical and sensing properties of nano-reinforced concrete made with coarse aggregates. The objective of this work was to fill this gap in the literature by evaluating the mechanical and self-sensing properties of CNF-reinforced concrete (CNFRC). Material tests were conducted on CNFRC cylinders and beams in order to determine the full compressive, tensile and flexural constitutive behaviour, including the post-peak response, as well evaluating the self-sensing capability. The results obtained are valuable for analysis and design of large critical infrastructures employing CNFRC

Some models of crack growth in brittle materials

This work is devoted to the study of models of fractures growth in brittle elastic materials; it collects the results obtained during my Ph.D., that are contained in [77, 76, 78]. We consider quasi-static rate-independent models, as well as rate-dependent ones and the case in which the first ones are limits of the second ones when certain physical parameters vanish. The term quasistatic means that, at each instant, the system is assumed to be in equilibrium with respect to its time-dependent data; this setting is typical of systems whose internal time scale is much smaller than that of the loadings. By rate-independent system we mean that, if the time-dependent data are rescaled by a strictly monotone increasing function, then the system reacts by rescaling the solutions in the same manner

Toughening Mechanisms in Silica-Filled Epoxy Nanocomposites

Epoxies are widely used as underfill resins throughout the microelectronics industry to mechanically couple and protect various components of flip-chip assemblies. Generally rigid materials largely surround underfill resins. Improving the mechanical and thermal properties of epoxy resins to better match those of their rigid counterparts can help extend the service lifetime of flip-chip assemblies. Recently, researchers have demonstrated that silica nanoparticles are effective toughening agents for lightly-crosslinked epoxies. Improvements in the fracture toughness of silica-filled epoxy nanocomposites have primarily been attributed to two toughening mechanisms: particle debonding with subsequent void growth and matrix shear banding. Various attempts have been made to model the contribution of these toughening mechanisms to the overall fracture energy observed in silica-filled epoxy nanocomposites. However, disparities still exist between experimental and modeled fracture energy results. In this dissertation, the thermal, rheological and mechanical behavior of eight different types of silica-filled epoxy nanocomposites was investigated. Each nanocomposite consisted of up to 10 vol% of silica nanoparticles with particle sizes ranging from 20 nm to 200 nm, with a variety of surface treatments and particle structures. Fractographical analysis was conducted with new experimental approaches in order to accurately identify morphological evidence for each proposed toughening mechanism. Overall, three major insights into the fracture behavior of real world silica-filled epoxy nanocomposites were established. First, microcracking was observed as an essential toughening mechanism in silica-filled epoxy nanocomposites. Microcracking was observed on the surface and subsurface of fractured samples in each type of silica-filled epoxy nanocomposite. The additional toughening contribution of microcracking to overall fracture energy yielded excellent agreement between experimental and modeled fracture energy results. Furthermore, the contribution of microcracking was most prevalent at lower filler contents which suggests that the presence of microcracking may account for the previously unexplained improvements in fracture behavior attained in silica-filled epoxy nanocomposites at low filler contents. Secondly, surface modification through the application of three different propriety surface treatments (“A”, “B” and “C”) was found to greatly influence the processibility and fracture behavior of silica-filled epoxy nanocomposites. B-treated silica nanoparticles were found to readily form micron-scale agglomerates, settled during nanocomposite curing and showed no improvement in fracture toughness with increasing filler content. In contrast, the nanocomposites consisting of A-treated and C-treated silica nanoparticles yielded morphologies primarily containing well-dispersed nanoparticles. Therefore, fracture toughness improved with increasing filler content. Finally, particle porosity was found to have no significant effect on fracture behavior for the range of silica-filled epoxy nanocomposites investigated. Lower density porous silica nanoparticles were just as effective toughening agents as higher density non-porous silica nanoparticles. Consequently, the potential exists for the use of toughened-epoxies in lightweight structural applications