A class of graphs with interesting structure is high-genusnear-planargraphs. These graphs have a highgenus, where the number of crossings would be ω(1), i.e., not bounded by a constant. Additionally, edge crossings would be present in multiple locations of the graph and thus would not be able to take advantage of constant number of topological handles or connections. Therefore these graphs are not able to be embedded in a surface of constant genus, so the genus of the graph would be not constant, i.e., ω(1), and clearly would not be planar. The near-planarity property of the graph would restrict the graph minimize non-planar crossing and provide structure that would allow an approach to tackling the separator theorem in this more general case. Edge crossings in near-planargraphs are limited so that the non-planar properties of the graph are restricted to local regions. To ensure this the crossed edges must be local to each other, i.e., the distance between the endpoints of crossed edge pairs is bounded by a constant. Additionally, this class of graphs does not necessarily have an excluded fixed minor. In the general case of high- genusnear-planargraphs, when attempting to contract vertices, the diagonal connections can be used to provide links that allow minors to be constructed. Therefore, these graphs cannot be characterized by the previous results about separators  .
To generalize our methods, it would be interesting to lift our results to more general graph classes, such as graphs with a fixed excluded minor. For Edge Multiway Cut, even the bounded-genus case remains open. Further work is also needed to improve the allowed error in Theorem 1.7. Currently, this error is an additive error of εw(∂B). In other words, a near- optimal Steiner forest is preserved only for “large” optimal forests, that is, for ones of size comparable to the perimeter of B. Is it possible to improve Theorem 1.7 to ensure a (1 + ε) multiplicative error? That is, to obtain a variant of Theorem 1.7 where the graph H satisfies w(F H ) ≤ (1 + ε)w(F B ), and thus to preserve near-optimal Steiner forests at all scales ? Finally,
dense graphs (i. e., of average degree at least 6 + ε , for some fixed ε > 0). For graphs of bounded degree, Chen, Kanchi, and Kanevsky  described a simple O( √ n)-approximation, which follows by the fact that graphs of small genus have small balanced vertex-separators. Following the present paper, Chekuri and Sidiropoulos  obtained a polynomial-time algorithm which, given a graph G of bounded degree and of genus g, outputs a drawing on a surface of genus O(g 12 log 19/2 n). Combined with the result of Chen et al., this implies a n 1/2−α approximation for bounded-degree graphs, for some constant α > 0. Subsequently, Kawarabayashi and Sidiropoulos  obtained a polynomial-time algorithm that computes a drawing on a surface of genus O(g 256 log 189 n) for general graphs (that is, without the condition that the maximum degree is bounded). Combined with Euler’s characteristic, this also implies a O(n 1−β )-approximation for general graphs, for some constant β > 0. We also remark that better bounds are known for 1-apex graphs [12, 8] (that is, graphs that can be made planar by removing a single vertex).
With the increase of the gammadion size, the distance between the individual gammadions in the array becomes smaller fostering the (increased) electromagnetic interaction between them. The optical ﬁeld distribution over the arrays of larger gammadions is even more complex than for weakly interacting gammadions (Fig. 5). The diﬀerence be- tween structures consisted of separated (Fig. 2) and closely packed gammadions (Fig. 5) is immediately seen: in the closely packed structure the strongest polarization conversion is observed near the area corresponding to the gammadion pattern symme- try points (like the centro-symmetrical area 1 of the pattern in Fig. 4). The areas corresponding to mutually perpendicular mirror-symmetrical points of the structure can be identiﬁed in the opti- cal distributions, however, in contrast to smaller gammadions discussed above, the relatively closed areas formed by the elements of four neighbouring
Thus, the induced current on the target’s surface plays the same role as the current source of an AUT. Proceeding from this, the Fourier transformation of the measured near-field gives the far-field scattering pattern. To convert the bistatic scattering pattern into bistatic RCS a calibration procedure is required .
Electrically Small Antennas (ESAs) are applied abundantly in modern communication systems [1–3]. A lot of methods have been studied to reduce the size of an antenna. Among others, this resulted in planar inverted-F antennas (PIFA), meander transmission line antennas loaded microstrip antennas, and surface mounted device (SMD) loaded antennas [4–9]. In the last decade, the use of metamaterials (MMs) also has shown to have potential to miniaturize antennas. It has been proven that a shell of ideal homogenous MM can reduce the antenna size, while keeping a high eﬃciency [10–15]. However, in practise a homogenous MM is hard to obtain, since the typical size of the MM unit cell is about 10% of the wavelength in vacuum.
Let’s look once again at the graph from the last chapter, in which we have one vertex per COPS participant and one edge between two participants when they have worked together in the past. If the COPS organizers end up defining their working groups in such a way that each participant has worked with exactly six other people in the group, then we know that the graph associated with a group is not planar, because all the vertices have a degree of 6.
The solution conﬁgurations that we consider are comprised of points in the plane with a number of speciﬁed distances between them. With the natural correspon- dence of points to vertices and constraint pairs to edges, each constraint system has an associated abstract graph. It is the nature of the abstract graph that is signiﬁ- cant for the solubility of the constraint system, and we shall be concerned with the situation where the abstract graph is a planar graph in the usual graph-theoretic sense; it can be drawn with edges realised by curves in the plane with no crossings. We show that a planar 3-connected maximally independent graph with generic distances is not only not solvable by quadratic extensions but is not soluble by radical extensions, that is, by means of the extraction of roots of arbitrary order together with the basic arithmetical operations. In fact our methods make use of some intricate planar graph theory leading to an edge contraction reduction scheme which is also of independent interest. The main theorem of the paper can be stated as follows.
However, some more care is needed to complete the proof: firstly, we need to subtract those graphs that are not 2-connected because one of the vertices of the complete graph is an endpoint of all connecting edges. There must be an edge between this vertex and all leaves, and there might be further edges between this vertex and other tree vertices, but no other edges connecting the tree and the complete graph. Hence the exponential generating function for such graphs is
technique from [ Fen15 ] , which handles easy cases such as connected components that are cycles and paths, contracts long paths, and applies slightly more technical rules to achieve a kernel. We then make use of a branching technique which yields bounded degree instances. The restrictive structure of path forests enables efficient branching. As part of branching, we delete any instances where more than k edges were removed, and then we pass the remaining instances to our application of the Cut&Count framework. Cut&Count [ Cyg11a ] is a general framework used for solving decision problems where there is a global constraint. It requires bounded treewidth and uses dynamic programming over a tree decomposition. The key idea is that when building potential solutions in the natural way via dynamic programming, the framework guarantees that non-path forests are seen an even number of times. Then we can count the number of times a potential solution is seen modulto two and obtain a guaranteed “no" if no solution exists and a “yes" with high probability if one does (this is via a one-sided Monte Carlo method). This naturally leads to our result in Theorem 1.
But there exists no polynomial-time algorithm for resource k-partitioning of graphs for k > 3. In this pa- per, we give a linear algorithm for ﬁnding a resource tripartition of a 3-connected planar graph based on a “nonseparating ear decomposition" of the given graph. “Nonseparating ear decomposition" is a generalization of “canonical decomposition" . “Canonical decom- position" is applied in convex grid drawing of planar graph . “Canonical decomposition" i.e. “canoni- cal ordering" has applications in producing straight line grid drawings with polynomial sizes for planargraphs. A “canonical decomposition", a “realizer", a “Schny- der labeling" and an “orderly spanning tree" of a plane graph play an important role in straight-line drawings, ﬂoorplanning, graph encoding etc. [2, 4, 6, 10, 12]. Miura et. al. proved that a “canonical decomposition", a “realizer", a “Schnyder labeling", an “orderly span- ning tree" and an “outer triangular convex grid draw- ing" are notions equivalent with each other . Hence
The proposed algorithm has a polynomial runtime; when using even a simple algorithm to find shortest paths, such as Dijkstra’s, the worst case run-time would be O (d(|E| + |V | log(|V |))), where d is the number of vertices in a shortest right-most path in the dual of the graph. This can be further simplified to O (d|V | log(|V |)) in simple planargraphs, as the number of edges in a simple planar graph is bounded by a constant multiple of |V | [Eul58]. While the algorithm is not restricted to simple graphs, and allows for counting minimum cuts in multigraphs, any multigraph in this context can be converted into a simple graph. Consider that any looping edge (u, u) will not contribute to a minimum cut weight, as u cannot be in both the cut set, and outside of the cut set, and so (u, u) can be preemptively removed from the graph G. Any duplicate edges e and f with endpoints (u, v) in the planar graph G, can be combined into a single edge x such that the weight of x is w(e) + w(f), as any cutset that includes u but not v would take the weight of both edges e and f . Finally, d is bounded by the number of vertices in G, simplifying the algorithms running time to O (|V | 2 lg(|V |))
Systems are often sold as “API type separators”, which commonly means that they have the same universal baffle arrangement as a regular API separator, but do not conform to the design criteria of a 45 minute residence time (established by the API). Separators which have a lesser residence time (and are therefore smaller and less expensive than rigorously designed API separators) do not meet the API design criteria and, therefore, cannot be expected to meet the API’s modest effluent expectations.
Several options exist for the design of a production target that can tolerate heavy-ion beam intensities in the range of 1 pµA. For solid target materials, such as C or Be, rotat- ing target wheels are well proven to be able to accomodate beam intensities of this magnitude. However, concerning gaseous targets such as deuterium, hydrogen, and helium, it is not possible to employ the present method with gas cells because of the fragility of the entrance and exit win- dows. Instead, we have developed a high intensity tar- get that generates a thin, liquid film by impinging two oil jets of ⇠ 0.5 mm diameter on each other inside a vacuum chamber. This mechanism generates a thin oil film of areal thickness ⇠ 1-2 mg / cm 2 needed for the primary target. The low vapor pressure of regular pump oil ensures that high vacuum can be maintained in the beam line. Several baf- fles and an in-line cold trap further ensure that the oil does not migrate along the beamline. The hydrogen content of the oil makes this target ideal for proton-induced reactions, e.g. (p,n) and (p,d). However, in order to use deuteron ini- tiated reactions, such as (d,n) and (d,p), a deuterated pump oil is required, an option that we’re presently researching.
to design spiral inductors at microwave frequencies using substrates such as Silicon (Si) [1–5] or gallium-arsenide (GaAs) [6–8]. A modeling technique using time domain (TD) impulse to characterize microwave spiral inductors is proposed in . LTCC spiral inductor design using the synthesis method is outlined in . Chip spiral inductor design which is mostly valid for CMOS processes is given in . In , design of the spiral inductors using layout synthesis and optimization technique on a crystalline polymer substrate is discussed. Nano-spiral inductor design for low power digital spintronic circuits is presented in . The inductance of the spiral inductor is calculated using Greenhouse-based formulation is given in [14–16]. Other methods such as empirical formulation in calculation of the inductance of the spirals are outlined in . The inductance of the spiral using a mathematical model with Kramers–Kronig relations are proposed in . There have been also reports for calculating the spiral inductor parameters based on the measured S-parameters [19, 20]. Numerical methods such as finite difference time domain method (FDTD) and multi resolution time domain method (MRTD) [21, 22] have been also applied for characterization of the spiral inductors. However, none of the publications reported presents an analytical model for the design of spiral inductors at HF (3 MHz–30 MHz) range for high power ISM applications. HF range is a common frequency range that is used for ISM (Industrial, scientific, and medical) applications. Spiral inductors, when used in ISM applications, must be designed to handle power in the range of several kilowatts. They should demonstrate good thermal characteristics, sustained inductance value and low loss under such high power because any change in the component values in RF system affects the performance and can cause catastrophic failures. This can be prevented by application of an accurate design method for the material that is used as a substrate.
It is worth noting here that both the algorithms of Elberfeld et al.  and Das et al.  cease to be polynomial time algorithms for classes of graphs whose treewidth w is not constant. These include a large number of interesting classes of graphs mentioned in the introduction. Our algorithm requires O(w log n) space and O(poly(n, w)) time and therefore has a better time and space complexity for solving the reachability problem when compared to the results of  and .
The tapered waveguide structure allows diode pumping at its multimode broad channel end and ensures fundamental-mode laser output at the single-mode channel end, with adiabatic operation achievable through careful design of the interconnecting taper. Linear and parabolic taper shapes are compared. The two types of waveguides expanding to various widths over the same lengths were fabricated on the same Nd:BK7 substrate and characterised with Ti:sapphire pumping. The linear tapers show superior operation for larger guiding sizes up to taper widths of 250μm, and therefore are more compatible with high-average-power broad-stripe diode pumping. Double-clad planar waveguides, fabricated by direct bonding YAG and sapphire, have features that are very attractive in this work: they are ideally suited to high- power diode bar/stack pumping owing to their high NA (0.46) slab-like geometry; and they are shown to robustly maintain single-mode operation by gain mode selection. Both diode bars and stacks were used to side-pump a 30μm double-clad Nd:YAG waveguide. For diode-bar pumping, an extended cavity was used to control the output spatial mode in the non-guided axis. Multimode output power larger than 10W was obtained from the waveguide with a slope efficiency of 56%, which was reduced to 33% when the external cavity was optimised for beam quality, obtaining
Although planarity is a severe restriction, we emphasize that planargraphs appear in many con- texts such as computer vision and image processing, magnetic and optical recording, or network routing and logistics. We have focused on inference problems defined on planargraphs with sym- metric pairwise interactions and, to make the problems difficult, we have introduced local field potentials. Notice however, that the algorithm can also be used to solve models with more complex interactions, that is, more than pairwise typical from the Ising model (see Chertkov et al., 2008, for a discussion of possible generalizations). This makes our approach more applicable than other ap- proaches, namely, (Globerson and Jaakkola, 2007; Schraudolph and Kamenetsky, 2008), designed specifically for the pairwise interaction case.
The presented algorithm for share assignment in communication networks uses node separators of the underlying graphs. This algorithm produces better results than simply placing large shares close to the user node, as is suggested in previous publications. One interesting question for further research is that under which assumptions concerning the underlying graph and set of shares, the heuristic presented here results in an optim- al placement of the shares on the nodes.
In this section, an eight-element L-band microstrip antenna array is constructed for measurement. The NSI near-ﬁeld measurement system is used. Fig. 4 shows photos of the experiment system and the power divider and phase shifters used. The proposed experiment system comprises: 1) a wooden support which has little eﬀect on the antenna performance; 2) an eight-way Wilkinson power divider; 3) coaxial cables with diﬀerent lengths to implement the four phase delays of ‘ ◦ ’, ‘90 ◦ ’, ‘180 ◦ ’ and ‘270 ◦ ’ at 2-GHz; 4) an eight-element microstrip antenna array at the resonant frequency of 2-GHz with 75 mm element spacing; 5) open-ended waveguide as the measurement probe. During the experimental study, the sinusoidal signal of 2 GHz generated by the vector network analyzer is equally divided into eight parts by the Wilkinson power divider. Diﬀerent phase delays of the eight-way signals are realized with the coaxial cables shown in Fig. 4(c). The delayed eight-way signals are radiated by the eight-element microstrip antenna array. The output signal of the open-ended waveguide is measured by the vector network analyzer.