Spanning Properties of Theta-Theta Graphs

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio 11.67, for k' >= 6. In this paper we establish a first spanning ratio of $7.82$ for Theta-Theta graphs, for the same values of $k$. We also extend the class of Theta-Theta spanners with parameter 6k', and establish a spanning ratio of $16.76$ for k' >= 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k' increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table

Tailoring tunnel magnetoresistance by ultrathin Cr and Co interlayers: A first-principles investigation of Fe/MgO/Fe junctions

We report on systematic ab-initio investigations of Co and Cr interlayers embedded in Fe(001)/MgO/Fe(001) magnetic tunnel junctions, focusing on the changes of the electronic structure and the transport properties with interlayer thickness. The results of spin-dependent ballistic transport calculations reveal options to specifically manipulate the tunnel magnetoresistance ratio. The resistance area products and the tunnel magnetoresistance ratios show a monotonous trend with distinct oscillations as a function of the Cr thickness. These modulations are directly addressed and interpreted by means of magnetic structures in the Cr films and by complex band structure effects. The characteristics for embedded Co interlayers are considerably influenced by interface resonances which are analyzed by the local electronic structure

Upward Point-Set Embeddability

We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph $D$ has an upward planar embedding into a point set $S$. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of $k$-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a $1$-switch tree), we show that not every $k$-switch tree admits an upward planar straight-line embedding into any convex point set, for any $k \geq 2$. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete

Metric Dimension for Gabriel Unit Disk Graphs is NP-Complete

We show that finding a minimal number of landmark nodes for a unique virtual addressing by hop-distances in wireless ad-hoc sensor networks is NP-complete even if the networks are unit disk graphs that contain only Gabriel edges. This problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The Gabriel edges of a unit disc graph induce a planar O(\sqrt{n}) distance and an optimal energy spanner. This is one of the most interesting restrictions of Metric Dimension in the context of wireless multi-hop networks.Comment: A brief announcement of this result has been published in the proceedings of ALGOSENSORS 201

Lower bounds on the dilation of plane spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6$, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2 table

An explanation for the rise in Tc in the Tl- and Bi-based high temperature superconductors

Using the plasmon exchange model for the high T(sub c) superconductor, it is shown that the T(sub c) rises with an increase in the number of CuO layers per unit cell, which is in agreement with recent observations in the Tl- and Bi-based compounds. The calculation also suggests that the sample will become superconducting in successive stages and that there is a saturation effect, i.e., that T(sub c) cannot be raised indefinitely by increasing the number of CuO layers

One pot copper catalyzed conversion of oximes to thioamides

Thioamides are important structural motifs found in many biologically active molecules and precursors for the synthesis of various fine chemicals, heterocycles etc. In view to the wide use of thioamides, development of a simple and practical method toward their synthesis is an active area of research. The classical methods for the preparation of thioamides usually involve the generation of amides followed by subsequent thionation using the thionating agents such as P4S10, Lawesson’s reagent, etc. Beckmann rearrangement is one of the most practised methods to produce amides. However, Beckmann rearrangement has some limitations such as the requirement of high reaction temperatures and the use of large amounts of strong Brønsted acids and dehydrating media followed by production of large amounts of byproducts. It has been observed that when aldoximes undergoing acid catalysed Beckmann rearrangement, it leads to nitriles, whereas in the presence of metal catalysts, primary amides are obtained. Usually, the transition metal-catalyzed reactions are proceeding through a dehydration/rehydration route via the formation of a discrete nitrile intermediate. A number of important transition metal (i.e. Rh, Ru, Ir, Pd, and Au/Ag) catalysts were successfully used to synthesize primary amides from aldoximes. Now, considering the low cost, low catalyst concentration, milder reaction condition and high abundance of Cu catalyst, makes it an attractive choice for reaction. Recently, Panda et.al from NIT Rourkela developed a Cu-catalyzed protocol for the regioselective synthesis of N-aryl amides from the reaction of aldoxime with aryl halide in moderate to good yield. In line with this work, we attempted and successfully developed a simple Cu-catalyzed direct synthesis of thioamides from aldoximes in one-pot in the presence of thionating agent. Here, this report discusses the simple Cu-catalysed protocol to synthesize thioamides from aldoximes in moderate to good yield

Spin systems with dimerized ground states

In view of the numerous examples in the literature it is attempted to outline a theory of Heisenberg spin systems possessing dimerized ground states (``DGS systems") which comprises all known examples. Whereas classical DGS systems can be completely characterized, it was only possible to provide necessary or sufficient conditions for the quantum case. First, for all DGS systems the interaction between the dimers must be balanced in a certain sense. Moreover, one can identify four special classes of DGS systems: (i) Uniform pyramids, (ii) systems close to isolated dimer systems, (iii) classical DGS systems, and (iv), in the case of $s=1/2$, systems of two dimers satisfying four inequalities. Geometrically, the set of all DGS systems may be visualized as a convex cone in the linear space of all exchange constants. Hence one can generate new examples of DGS systems by positive linear combinations of examples from the above four classes.Comment: With corrections of proposition 4 and other minor change