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Topological extensions of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Let ≺ be a total order on edges E(G) of a ribbon graph G, and Q be a spanning quasi-tree of G. Tracing the boundary component of Q yields a round trip passing the boundary arcs of each edge-ribbon twice. These two boundary arcs will intersect faces for edge in Q, and will intersect vertices for edges not in Q. This information can be encoded in a chord diagram CG(Q) consisting of a circle corresponding to the boundary of Q with edge arcs recorded in the cyclic order they appear on the boundary of Q, and with pairs corresponding to the same edge-ribbon connected by chords. Thus the set of chords inherits the total order ≺. Figure 27.5(b) shows the chord diagram corresponding to the quasi-tree on {a,b,c} in the graph from Example 27.13.
Fields in Hollow Rectangular Waveguides
Published in Philip C. Magnusson, Gerald C. Alexander, Vijai K. Tripathi, Andreas Weisshaar, Transmission Lines and Wave Propagation, 2017
Philip C. Magnusson, Gerald C. Alexander, Vijai K. Tripathi, Andreas Weisshaar
A comparison of the results in Eqs. 13-39 and 13-40 with those in Chap. 3 for two lines in tandem (Eqs. 3-37 and 3-39) suggests that, for a consistent analogy, waveguide characteristic impedance (for the TE10 mode) should be defined as inversely proportional to β10. Perhaps the simplest basis for defining absolute impedances in waveguides, a definition which meets this requirement and which may be applied consistently to other modes, is the following: Let Etrv(z)aEt and Htrv(z)aHt represent the phasor forms of the transverse components of the respective fields, with the positive directions of the unit vectors so chosen that aEt, aHt, and the direction to the load (in that cyclic order) form a right-hand set. Then let
Graphs and Surfaces
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
A closed surface is a generalized notion of a polyhedron. A polyhedron is a three-dimensional object consisting of a set of polygons, of three or more sides each. Each polygon is bounded by a sequence of p straight-line segments connecting p vertices in cyclic order, for some p ≤ 3. The line segments are the edges of the polyhedron. Each edge is shared by exactly two polygons. Any two polygons may intersect only on a single common edge. There are at least three polygons meeting at each vertex, and the polygons meeting at any vertex form a single cycle.
Linear convergence of proximal incremental aggregated gradient method for nonconvex nonsmooth minimization problems
Published in Applicable Analysis, 2022
Unlike most of the existing PG using the full gradient of f, PIAG uses an aggregated gradient at each iteration k. In particular, if the delay , i.e. , then , PIAG reduces to PG. In addition, when the nonconvex regularization function g vanishes, if with initialization for all i, and admits the recursion i.e. the component functions are processed one by one using a deterministic cyclic order on the index set . This is the original IAG method introduced by Blatt et al. in [26].
A reconstruction of object properties with significant uncertainties
Published in Inverse Problems in Science and Engineering, 2021
Having proposed the restriction of the desired solutions in general by the scheme (6), we will implement the relaxation rules for solutions to ill-posed problems. The reconstruction should be based on the elementary step, which consists of varying only one element of the set of unknowns, and the number of variable elements being chosen in a specific cyclic order. The restriction technique described below is the essential addition to the stabilizer’s implementation because local disturbances of a monotonous behaviour will be bounded for the cases, where the stabilizer cannot exclude the appearance of spikes.
Mathematical models and routing algorithms for economical cutting tool paths
Published in International Journal of Production Research, 2018
T.A. Makarovskikh, A.V. Panyukov, E.A. Savitskiy
While constructing control manipulating systems using graphs one may map the different elements of manipulator trajectory. Here routing problems with different restrictions arise. For example, routes in which the next edge is defined by the given cyclic order on the set of edges incident to a current vertex (Fleischner 1991; Beineke, Fleischner, and Wilson 1983); routes in which some edges should be passed in a predefined order (Beineke, Fleischner, and Wilson 1983); or routes satisfying the restriction of ordered enclosing (OE-routes) (Panyukova and Panyukov 2000).