Miniaturization of a Koch-Type Fractal Antenna for Wi-Fi Applications

Koch-type wire dipole antennas are considered herein. In the case of a first-order prefractal, such antennas differ from a Koch-type dipole by the position of the central vertex of the dipole arm. Earlier, we investigated the dependence of the base frequency for different antenna scales for an arm in the form of a first-order prefractal. In this paper, dipoles for second-order prefractals are considered. The dependence of the base frequency and the reflection coefficient on the dipole wire length and scale is analyzed. It is shown that it is possible to distinguish a family of antennas operating at a given (identical) base frequency. The same length of a Koch-type curve can be obtained with different coordinates of the central vertex. This allows for obtaining numerous antennas with various scales and geometries of the arm. An algorithm for obtaining small antennas for Wi-Fi applications is proposed. Two antennas were obtained: an antenna with the smallest linear dimensions and a minimum antenna for a given reflection coefficient.

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε → 0 . The effective Hamiltonian depends on the spectral properties of an auxiliary ε -independent Hamiltonian defined on the compact graph obtained by setting ε = 1 . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε → 0 , the leads are decoupled.

Some families of 0-rotatable graceful caterpillars

A graceful labelling of a tree T is an injective function f: V (T) → {0, 1, . . . , |E(T)|} such that {|f(u)−f(v)|: uv ∈ E(T)} = {1, 2, . . . , |E(T)|}. A tree T is said to be 0-rotatable if, for any v ∈ V (T), there exists a graceful labelling f of T such that f(v) = 0. In this work, it is proved that the follow- ing families of caterpillars are 0-rotatable: caterpillars with perfect matching; caterpillars obtained by identifying a central vertex of a path Pn with a vertex of K2; caterpillars obtained by identifying one leaf of the star K1,s−1 to a leaf of Pn, with n ≥ 4 and s ≥ ⌈n−1 2 ⌉; caterpillars with diameter five or six; and some families of caterpillars with diameter at least seven. This result reinforces the conjecture that all caterpillars with diameter at least five are 0-rotatable.

All Ramsey Numbers for Brooms in Graphs

For $k,\ell\ge 1$, a broom $B_{k,\ell}$ is a tree on $n=k+\ell$ vertices obtained by connecting the central vertex of a star $K_{1,k}$ with an end-vertex of a path on $\ell-1$ vertices. As $B_{n-2,2}$ is a star and $B_{1,n-1}$ is a path, their Ramsey number have been determined among rarely known $R(T_n)$ of trees $T_n$ of order $n$. Erdős, Faudree, Rousseau and Schelp determined the value of $R(B_{k,\ell})$ for $\ell\ge 2k\geq2$. We shall determine all other $R(B_{k,\ell})$ in this paper, which says that, for fixed $n$, $R(B_{n-\ell,\ell})$ decreases first on $1\le\ell \le 2n/3$ from $2n-2$ or $2n-3$ to $\lceil\frac{4n}{3}\rceil-1$, and then it increases on $2n/3 < \ell\leq n$ from $\lceil\frac{4n}{3}\rceil-1$ to $\lfloor\frac{3n}{2}\rfloor -1$. Hence $R(B_{n-\ell,\ell})$ may attain the maximum and minimum values of $R(T_n)$ as $\ell$ varies.

Force–Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms

The origami waterbomb base (WB) is a single-vertex bistable mechanism that can be generalized to accommodate various geometric, kinematic, and kinetic needs. The traditional WB consists of a square sheet that has four mountain folds alternating with five valley folds (eight folds total) around the vertex in the center of the sheet. This special case mechanism can be generalized to create two classes of waterbomb-base-type mechanisms that allow greater flexibility for potential application. The generalized WB maintains the pattern of alternating mountain and valley folds around a central vertex but it is not restricted to eight total folds. The split-fold waterbomb base (SFWB) is made by splitting each fold of a general WB into two “half folds” of the same variety as the parent fold. This study develops kinematic, potential energy, and force–deflection models for the rigid-foldable, developable, symmetric cases of the generalized WB and the SFWB, and investigates the relative effects of numbers of folds and split-fold panel size, on device behavior. The effect of selected key parameters is evaluated, and equations are provided to enable the exploration of other important parameters that may be of interest in the design and analysis of specific mechanisms. The similarities and differences between the two general forms are discussed, including tunability of the bistable and force–deflection behavior of each.

The Class of q-Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs, and an Entomological Experiment

Much research has involved the consideration of graphs which have subgraphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, that is, they have a pair a, b of nonzero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define a new class of graphs, called q-cliqued graphs, on q2+1 vertices, which contain q cliques each of order q connected to a central vertex, and then prove that these q-cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is -1 and product 1-q. These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment.

FOLDING EQUILATERAL PLANE GRAPHS

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

APPROXIMATING THE SPANNING k-TREE FOREST PROBLEM

The spanning star forest problem is an interesting algorithmic problem in combinatorial optimization and finds different applications. We generalize it into the spanning k-tree forest problem, which is to find a maximum spanning forest in which each tree component has a central vertex and other vertices in the component have distance at most k away from the central vertex. We show that this new problem can be approximated with ratio [Formula: see text] in polynomial time for both undirected and directed graphs. In the weighted distance model, a ½-approximation algorithm is presented for it.