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Homotopy Algorithms for Engineering Analysis
Published in Hojjat Adeli, Supercomputing in Engineering Analysis, 2020
Layne T. Watson, Manohar P. Kamat
has full rank (rank = 3) on ρ−1(0). In differential geometry jargon, ρ is said to be transversal to zero. The mathematics then says that for almost all vectors d ∈ E3 (in the sense of Lebesgue measure), the map ρd is also transversal to zero. What this means geometrically is that the zero set of ρd consists of smooth disjoint curves that do not intersect themselves or bifurcate, and have endpoints only at λ = 0 or λ = 1 (see Figure 1). In general under suitable conditions (described in Section 2) there is a zero curve γ of ρd(λ,v) stretching from a known solution v0 at λ = 0 to the desired solution v¯ at λ = 1.
Preliminaries
Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020
Aliakbar Montazer Haghighi, Indika Wickramasinghe
To state the AC, we choose the relatively simple version used by Russell who referred to it as the multiplicative axiom. The AC states that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set, referred to as a transversal or choice set. This set is to contain exactly one element from each member of the given collection. Symbolically, let C be a nonempty set. Then, there is a function, say f, defined as f:C⇒∪C∍(∀A∈C)[f(A)∈A].
Effect of lateral confining condition of behaviour of confined-reinforced earth
Published in Andrew McNamara, Sam Divall, Richard Goodey, Neil Taylor, Sarah Stallebrass, Jignasha Panchal, Physical Modelling in Geotechnics, 2018
Two strain gauges were attached in pairs on both sides of each measuring point, i.e., on the upper and lower surfaces of the geogrid. The tie rods were set with spacing of 400 mm in the longitudinal direction and 200 mm in the transversal direction. They were placed at 200 mm, 600 mm and 1000 mm from the left to right of the soil box. Overburden on the CRE was 4 kPa to simulate weight of a pavement. Therefore, the overburden on the geogrids (G1-4) were 4, 5.5, 7, and 8.5 kPa, respectively (weight of each sand layer was 1.5 kPa). These values were smaller than those used in the field because the size of the CRE model in the laboratory was smaller. It should be noted that in order to keep the sand from leaking when the CRE deformed, the sand layers were wrapped in a thin low strength geotextile.
Extended formulations of lower-truncated transversal polymatroids
Published in Optimization Methods and Software, 2021
Hiroshi Imai, Keiko Imai, Hidefumi Hiraishi
Martin [9] firstly shows a class of matroids with polynomial extended formulations by reformulating problems with new auxiliary variables as follows. The base polytope of a graphic matroid, for a graph with vertex set V and edge set E, is shown to have an extended formulation of size ( as pointed out in [2]). The base polytope of a transversal matroids on U, over a bipartite graph with left vertex set U, right vertex set W and edge set , is shown to have an extended formulation of size . Iwata et al. [8] show the base polytope of a -sparsity matroid on has extension complexity of when , and when k<l by devising randomized communication protocols as an extension of the protocol in Faenza et al. [2]. For bipartite matchings, the Birkhoff polytope on perfect matchings gives a polynomial-size extended formulation directly [3], which directly implies the above result of transversal matroids.
A stochastic approach to the closure of accessible sets of control systems with application on homogeneous spaces
Published in Dynamical Systems, 2018
Diego S. Ledesma, Fabiano B. da Silva
Now, assume that μ1 is another harmonic probability measure such that its support is a closed saturated set K⊊M. The Radon–Nikodym derivative dμ1/dμ = f is a leafwise harmonic function on each leaf L⊂K; in fact, Therefore, f is constant on μ almost each L (Garnett [7, Thm. 1.b]). Let L0⊂K be a leaf such that . Let T be an open transversal section to L0 and consider the saturated set Each leaf through T will intercept T at a single point t. Denote by Lt to the leaf such that T∩Lt = {t}. Let gt ∈ G be such that gt · Lt = L0; moreover, the map defined by defines an isometry and therefore Lt and L0 are isometric.
The inventory of structural typologies of timber floor slabs and roofs in the monumental built heritage: the case of the Royal Palace of Naples
Published in International Journal of Architectural Heritage, 2018
Beatrice Faggiano, Anna Marzo, Maria Rosaria Grippa, Giacomo Iovane, Federico M. Mazzolani, Donato Calicchio
The on-site surveys have evidenced that the primary beams have generally circular cross sections with variable diameter along the axis. Spans are large, therefore beams are generally stiffened by inclined struts and strengthened, in the middle part, through longitudinal secondary beams, in all realizing in plane portal frames (Figure 13). In some cases, also transversal beams are set up. Moreover, at the end parts of the beam, at the support to the masonry walls, cantilevered timber thick planks, with rectangular cross section, are placed at both sides, as strengthening systems, increasing the cross section (Figure 13). In some cases, beams are stiffened by transversal inclined struts arranged in planes perpendicular to the primary beam (Figure 13).